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\title{
\rule{\linewidth}{1pt} \\
\textbf{
    Automorphic Forms Beyond GL2
  }
    \\ {\normalsize Arizona Winter School 2022 Lectures} \\
  \rule{\linewidth}{2pt}
}
\titlehead{
    \begin{center}
  \includegraphics[width=\linewidth,height=0.45\textheight,keepaspectratio]{figures/cover.png}
  \end{center}
       \begin{minipage}{.35\linewidth}
    \begin{flushleft}
      \vspace{2em}
      {\fontsize{6pt}{2pt} \textit{Notes: Sketchy live-tex'd notes, any
errors are my own! } } \\
    \end{flushleft}
    \end{minipage}
    \hfill
    \begin{minipage}{.65\linewidth}
    \end{minipage}
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2022-03-09 }
\end{flushleft}


\newpage

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\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{plenary-akshay-venkatesh}{%
\section{Plenary: Akshay Venkatesh}\label{plenary-akshay-venkatesh}}

\begin{remark}

A fairy tale: see
\href{https://www.youtube.com/watch?v=8yRNQIkiudQ}{Langlands Elephant}.
The point of this talk is to see the entire elephant!

\end{remark}

\begin{remark}

In the analogy between number fields and 3-manifolds, automorphic forms
are on the number field side -- what is the manifold analog? Some
history:

\begin{itemize}
\tightlist
\item
  Mazur 63/64, in conversations with Artin:
  \(\operatorname{Spec}{\mathbb{Z}}/p{\mathbb{Z}}\to \operatorname{Spec}{\mathbb{Z}}\)
  is like a knot in a (simply connected) 3-manifold.
\item
  Weil 49: Weil conjectures, there should be an ``algebraic'' cohomology
  theory \(H^*({-})\) for \(X_{/ {k}}\) for
  \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
  where \(H^*(X({\mathbb{C}}))\) recovers singular cohomology.

  \begin{itemize}
  \tightlist
  \item
    Artin/Grothendieck: for finite coefficients, realized this using
    \textbf{étale cohomology} \(H^*_\text{ét}(X)\).
  \end{itemize}
\item
  Tate 62, Poitou 61: looked at Galois cohomology, showed
  \(H_\text{ét}^*(\operatorname{Spec}{\mathbb{Z}})\) (or other number
  rings) has a ``Poincare duality'' \(H^i \rightleftharpoons H^{3-i}\),
  making it look like a 3-manifold. Recovers results in class field
  theory.
\end{itemize}

\end{remark}

\begin{example}[Varieties with automorphisms]

Let \(X = V(x^3+y^3+z^3) \subseteq {\mathbb{P}}^2_{/ {{\mathbb{C}}}}\);
there are automorphisms

\begin{itemize}
\tightlist
\item
  \(x\mapsto y, x\mapsto z\), etc
\item
  \(x\mapsto \mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu, y\mapsto \mkern 1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern 1.5mu\),
  etc
\item
  \(x,y,z \mapsto \sigma(x), \sigma(y), \sigma(z)\) for
  \(\sigma\in S_3\)
\end{itemize}

These should all act on \(H^*(X)\) for any such \(H^*\), but conjugation
is quite discontinuous.

\end{example}

\begin{example}[Manifolds vs schemes]

Let \(X \in {\mathsf{Mfd}}\), how does one compute
\(H^*_{\mathrm{sing}}(X; {\mathbb{Z}})\)? Reduce to a linear-algebraic
problem by triangulating and forming a chain complex of simplices.
However, for
\(X \coloneqq\operatorname{Spec}R, R\coloneqq{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{2} } \right] }\),
\begin{align*}
H^1_\text{ét}(X; C_2) &= R^{\times}/{ {R^{\times}}_{\scriptscriptstyle \square} }= \left\{{\pm 1,\pm 2}\right\} \cong C_2{ {}^{ \scriptscriptstyle\times^{2} }  } \\
H^2_\text{ét}(X; C_2) &= {\mathsf{Quat}}{\mathsf{Alg}}_{/ {R}}  = \left\{{ \operatorname{Mat}_{2\times 2}(R), R[i,j,k]/\left\langle{i^2=j^2=k^2=-1}\right\rangle }\right\}
.\end{align*}
so \(H^2\) classifies quaternion algebras over \(R\). So computing this
is very different to the case of manifolds! Also note that these are not
dual on-the-nose, since \(H^1,H^2\) have different orders.

\end{example}

\begin{remark}

Comparing duality for number rings vs 3-manifolds. For us, a number ring
will be

\begin{itemize}
\tightlist
\item
  \({\mathcal{O}}_{K, S}\) for \(K\) a number field, where \(S\) is a
  finite number of primes we can invert (the
  \textbf{\(S{\hbox{-}}\)integers of \(K\)}), or
\item
  functions on a smooth curve over \({\mathbb{F}}_q\).
\end{itemize}

For simplicity, we'll take
\(R = {\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }, X = \operatorname{Spec}R\).
For \(M\in{\mathsf{Ab}}{\mathsf{Grp}}\) a \(p{\hbox{-}}\)torsion group
(where we need the order to be invertible in \(R\)), we'll consider
\begin{align*}
H^i(X; M) \coloneqq H^i_\text{ét}(\operatorname{Spec}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }; M)
.\end{align*}
There is a LES involving
\(M {}^{ \vee }\coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Grp}}(M, S^1) = \mathop{\mathrm{Hom}}_{\mathsf{Grp}}(M, \mu_{p^\infty})\)
where \(\mu_{p^\infty}\) are the \(p{\hbox{-}}\)power roots of 1:

\begin{center}
\begin{tikzcd}
    &&&& {H^{i-1}(\cdots)} \\
    \\
    {H^{3-i}({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }; M {}^{ \vee }) {}^{ \vee }} && {H^i({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }; M)} && {H^i({ {\mathbb{Q}}_p }; M)} \\
    \\
    {H^{i+1}(\cdots)}
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[from=1-5, to=3-1]
    \arrow[from=3-5, to=5-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMiwyLCJIXmkoXFxaWlxcaW52ZXJ0IHA7IE0pIl0sWzQsMiwiSF5pKFxcUVFwYWRpYzsgTSkiXSxbMCwyLCJIXnszLWl9KFxcWlpcXGludmVydCBwOyBNXFxkdWFsKVxcZHVhbCJdLFs0LDAsIkhee2ktMX0oXFxjZG90cykiXSxbMCw0LCJIXntpKzF9KFxcY2RvdHMpIl0sWzIsMF0sWzAsMV0sWzMsMl0sWzEsNF1d}{Link
to Diagram}
\end{quote}

On the other hand, let \(X\in {\mathsf{Mfd}}^3\) be a manifold with
boundary, then there is a LES

\begin{center}
\begin{tikzcd}
    &&&& {H^{i-1}(\cdots)} \\
    \\
    {H^i(X, {{\partial}}X; M) \cong H^{3-i}(X; M {}^{ \vee }) {}^{ \vee }} && {H^i(X; M)} && {H^i({{\partial}}X; M)} \\
    \\
    {H^{i+1}(\cdots)}
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[from=1-5, to=3-1]
    \arrow[from=3-5, to=5-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMiwyLCJIXmkoWDsgTSkiXSxbNCwyLCJIXmkoXFxiZCBYOyBNKSJdLFswLDIsIkheaShYLCBcXGJkIFg7IE0pIFxcY29uZyBIXnszLWl9KFg7IE1cXGR1YWwpXFxkdWFsIl0sWzQsMCwiSF57aS0xfShcXGNkb3RzKSJdLFswLDQsIkhee2krMX0oXFxjZG90cykiXSxbMiwwXSxbMCwxXSxbMywyXSxbMSw0XV0=}{Link
to Diagram}
\end{quote}

So
\(X \approx {\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\)
is a (nonorientable!) 3-manifold in this analogy, with
\({{\partial}}X \approx { {\mathbb{Q}}_p }\), which is now like a
2-manifold. The nonorientable assumption here is related to the need to
twist the Galois actions on the scheme side. How to realize this: delete
a tubular neighborhood of the knot, then

\begin{itemize}
\tightlist
\item
  \({{\partial}}X = {{\partial}}\nu(\operatorname{Spec}{\mathbb{Z}}/p{\mathbb{Z}}) \approx { {\mathbb{Q}}_p }\).
\item
  \(\operatorname{Spec}{ {\mathbb{Z}}_p }\approx \nu\),
\item
  \({{\partial}}\operatorname{Spec}{ {\mathbb{Z}}_p }\approx {{\partial}}\operatorname{Spec}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] } \sim \operatorname{Spec}{ {\mathbb{Q}}_p }\).
\end{itemize}

\includegraphics{figures/2022-03-05_11-44-31.png}

\end{remark}

\begin{remark}

\envlist

\begin{itemize}
\tightlist
\item
  The 3-dimensional objects: \(\operatorname{Spec}R\) for
  \(R = {\mathbb{Z}}, {\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }, {\mathbb{Z}} { \left[ \scriptstyle {\sqrt 2} \right] }\),
  or \({\mathbb{F}}_p(t)\), \({ {\mathbb{Z}}_p }\), or projective smooth
  curves over \({\mathbb{F}}_p\).
\item
  The 2-dimensional objects:
  \({ {\mathbb{Q}}_p }, {\mathbb{F}}_p{\left(\left( t \right)\right) }\),
  or projective smooth curves over
  \(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\).
\end{itemize}

\end{remark}

\begin{remark}

Relating to automorphic forms: for \(G={\operatorname{SL}}_2\),

\begin{itemize}
\tightlist
\item
  For \({\mathbb{Z}}\), we study the vector space
  \({\mathcal{A}}_{\mathbb{Z}}= \left\{{\text{functions on } \dcosetl{G({\mathbb{Z}})}{G({\mathbb{R}})}}\right\}\).
\item
  For \({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\),
  we instead look at
  \({\mathcal{A}}_{{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }}= \left\{{\text{functions on } \dcosetl{G({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] })}{ G({\mathbb{R}}) \times G({ {\mathbb{Q}}_p })} }\right\}\).
\end{itemize}

We would like some association that works similarly:
\begin{align*}
{\mathsf{Mfd}}^3 &\to { \mathsf{Vect} }_{/ {k}}  \\
M &\mapsto {\mathcal{A}}_M
.\end{align*}

\end{remark}

\begin{example}[?]

A non-example is \(M\mapsto H_{\mathrm{sing}}^*(M; {\mathbb{C}})\),
which behaves nothing like
\({\mathbb{Z}}\mapsto {\mathcal{A}}_{\mathbb{Z}}\). E.g. it has the
wrong type of functoriality: the map
\({\mathbb{Z}}\to {\mathbb{Z}} { \left[ \scriptstyle {\sqrt 2} \right] }\)
is like a branched double cover, but for manifolds there are maps both
ways and here it is difficult to find wrong-way maps. Moreover the
corresponding spaces of automorphic forms
\({\mathcal{A}}_{{\mathbb{Z}}}, {\mathcal{A}}_{{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }}\)
differ by far more than just a part coming from \({ {\mathbb{Q}}_p }\).

Note also that \(H^*(M \cup N) \cong H^*(M) \oplus H^*(N)\), but for
automorphic forms we have
\({\mathcal{A}}_{{\mathbb{Z}}\oplus {\mathbb{Z}}} \cong {\mathcal{A}}_{\mathbb{Z}}\otimes{\mathcal{A}}_{\mathbb{Z}}\),
where the source is like a degenerate quadratic extension.

\end{example}

\begin{remark}

A heavily studied piece of the analogy:
\begin{align*}
{\operatorname{ {\text{Automorphic forms}} }}\rightleftharpoons{\mathsf{TQFT}}
,\end{align*}
coming from the
\href{https://arxiv.org/abs/hep-th/0604151}{Kapustin-Witten 2006}, where
a (4-dimensional) TQFT is a essentially a monoidal functor:
\begin{align*}
{\mathsf{TQFT}}^3 = [({\mathsf{Bord}}^3, {\textstyle\coprod}) \to ({ \mathsf{Vect} }_{/ {k}} , \otimes_k)]
.\end{align*}
.

\begin{quote}
See \href{https://www.math.ru.nl/~mueger/TQFT/At.pdf}{Atiyah TQFT},
section 2.
\end{quote}

\end{remark}

\hypertarget{closing-akshay-venkatesh}{%
\section{Closing: Akshay Venkatesh}\label{closing-akshay-venkatesh}}

\begin{remark}

Automorphic forms using TFQTs as a metaphor. We'll consider
\({\mathsf{TQFT}}_4 = [({\mathsf{Bord}}^3, {\textstyle\coprod}), ({ \mathsf{Vect} }_{/ {{\mathbb{C}}}} , \otimes_{\mathbb{C}})]\)
where \({\mathsf{Bord}}^3\) is the category whose objects are
3-manifolds and morphisms \(M\to N\) are 4-manifolds \(W\) with
\({{\partial}}W = M {\textstyle\coprod}N\). These are meant to extract
invariants of 4-manifolds that are amenable to cut-and-paste arguments.
The correspondences:

\begin{itemize}
\tightlist
\item
  Manifolds \(M \leadsto\) vector spaces \(A_M\),
\item
  Bordisms \((M\to N)\leadsto\) linear maps \(A_M\to A_N\),
\item
  A 4-manifold \(Z\) with boundary \(M \leadsto\) a vector
  \(v_Z \in A_M\)
\item
  A 4-manifold \(Z\) with empty boundary
  \(\leadsto \lambda_Z\in {\mathbb{C}}\)
\item
  A decomposition \(X\) without boundary into
  \(M{\textstyle\coprod}_Z N \leadsto\) vector
  \(\ell\in A_M, r\in A_M {}^{ \vee }\), where
  \({\left\langle {\ell},~{r} \right\rangle} = \lambda_Z\).
\end{itemize}

\end{remark}

\begin{example}[A $\TQFT_2$ due to Dijkgraaf-Witten]

Fix \(G\in{\mathsf{Fin}}{\mathsf{Grp}}\), and a correspondence

\begin{itemize}
\tightlist
\item
  \(S^1 \leadsto {\mathbb{C}}[G]^{\mathrm{conj} }\), conjugacy-invariant
  functions in the group algebra,
\item
  Pants \(\leadsto\) multiplication
\item
  \(\Sigma_g\) a genus \(g\) surface \(\leadsto {1\over k!} N\) where
  \(N\) is the number of ways to write \(e\in G\) as a product of \(g\)
  commutators.
\end{itemize}

\end{example}

\begin{remark}

An informal definition of \textbf{extended TQFTs}, in particular
\({\mathsf{TQFT}}_4\):

\begin{itemize}
\tightlist
\item
  4-manifolds \(\leadsto {\mathbb{C}}\)
\item
  3-manifolds \(\leadsto { \mathsf{Vect} }_{/ {{\mathbb{C}}}}\)
\item
  2-manifolds \(\leadsto\) categories enriched over
  \({ \mathsf{Vect} }_{/ {{\mathbb{C}}}}\)
\end{itemize}

This should yield

\begin{itemize}
\tightlist
\item
  3-manifolds without boundary \(\leadsto A_S\) objects in
  \(\mathsf{C}\)
\item
  \(X = M{\textstyle\coprod}_Z N \leadsto \mathop{\mathrm{Hom}}(A_M, A_N)\)
\end{itemize}

\end{remark}

\begin{remark}

Last time we said
\(\operatorname{Spec}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\)
is like a 3-manifold with boundary \({ {\mathbb{Q}}_p }\), which is like
a 2-manifold. A philosophy is to put all places on the same footing --
note that we haven't included the places at \(p\) and \(\infty\) here,
so really we should have
\({{\partial}}\operatorname{Spec}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] } = \operatorname{Spec}{\mathbb{R}}, \operatorname{Spec}{ {\mathbb{Q}}_p }\),
and
\({{\partial}}\operatorname{Spec}{\mathbb{Z}}= \operatorname{Spec}{\mathbb{R}}\).
So our new picture should be:

(todo finish)

\end{remark}

\hypertarget{automorphic-forms-as-mathsftqft_4}{%
\subsection{\texorpdfstring{Automorphic Forms as
\({\mathsf{TQFT}}_4\)}{Automorphic Forms as \{\textbackslash mathsf\{TQFT\}\}\_4}}\label{automorphic-forms-as-mathsftqft_4}}

\begin{remark}

What should the automorphic form correspondence be in this analogy? Our
3-dimensional objects:

\begin{itemize}
\tightlist
\item
  \({\mathbb{Z}}\leadsto {\mathcal{A}}_{\mathbb{Z}}\) functions on
  \(\dcosetl{G_{\mathbb{Z}}}{G_{\mathbb{R}}}\),
\item
  \({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\leadsto {\mathcal{A}}_{{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }}\)
  functions on
  \(\dcosetl{?}{G_{\mathbb{R}}{ {}^{ \scriptscriptstyle\times^{2} } }}\),
\item
  \(X_{/ {{\mathbb{F}}_p}}\) a smooth projective curve \(\leadsto\)
  functions on \({\mathsf{Bun}}_G(X)\),
\item
  \({ {\mathbb{Z}}_p }\leadsto\) something more difficult.
\end{itemize}

The 2-dimensional objects:

\begin{itemize}
\tightlist
\item
  \({ {\mathbb{Q}}_p }\leadsto {\mathsf{Rep}}G({ {\mathbb{Q}}_p })\),
\item
  \({\mathbb{R}}\leadsto {\mathsf{Rep}}G({\mathbb{R}})\),
\item
  \(X_{/ {\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p}}\)
  a smooth projective curve
  \(\leadsto {\mathsf{Sh}}({\mathsf{Bun}}_G(X))\).
\end{itemize}

Really for a TQFT, we should assign something to objects in the category
of its boundary -- here e.g.~the vector space
\({\mathcal{A}}_{{\mathbb{Z}}}\) is an object in
\({\mathsf{Rep}}G({ {\mathbb{Q}}_p })\). Idea: functions on
\({\mathsf{Bun}}_G\) are hard to deal with, e.g.~Hecke operators turn
into infinite sums. Make things robust to passing to algebraic closures
by passing from functions to sheaves!

\end{remark}

\begin{remark}

Last time: thinking of \(\operatorname{Spec}{\mathbb{Z}}\) as a
3-manifold,
\begin{align*}
\operatorname{Spec}{\mathbb{Z}}= \operatorname{Spec}{ {\mathbb{Z}}_p }{ \displaystyle\coprod_{\operatorname{Spec}{ {\mathbb{Q}}_p }} } \operatorname{Spec}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] } \approx M^3{ \displaystyle\coprod_{Z^2} }N^3
.\end{align*}
We want the following:
\begin{align*}
\mathop{\mathrm{Hom}}_{G({ {\mathbb{Q}}_p })}({\mathcal{A}}_{{ {\mathbb{Z}}_p }}, {\mathcal{A}}_{{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }}) =_? {\mathcal{A}}_{{\mathbb{Z}}}
.\end{align*}

We regard \({\mathcal{A}}_{\mathbb{Z}}\) as elements of
\({\mathcal{A}}_{{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }}\)
which are also unramified at \(p\), and by Frobenius reciprocity this
should yield
\begin{align*}
{\mathcal{A}}_{\mathbb{Z}}= \mathop{\mathrm{Hom}}_{G({ {\mathbb{Q}}_p })}
\qty{\left\{{
\text{Functions on } \dcosetr{ G({ {\mathbb{Q}}_p })}{G({ {\mathbb{Z}}_p })} , {\mathcal{A}}_{{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }}
}\right\}}
,\end{align*}
which should encode a Hecke algebra at \(p\).

\end{remark}

\begin{question}

What is the Langlands correspondence in this language? What should an
``arithmetic TQFT'' be?

\end{question}

\begin{remark}

Let \(O\) be a category of arithmetic ring and \(A_O\) be a category or
vector space, and call an associated \(O\to A_O\) an \emph{arithmetic
field theory}. Let \(X_{/ {{\mathbb{F}}_p}}\) be a smooth projective
curve and \(G=\operatorname{GL}_n\). The Langlands correspondence here
(due to Drinfeld and Lafforgue) yields
\begin{align*}
\left\{{\substack{
  \text{Cuspidal functions on $n{\hbox{-}}$dimensional} \\
  \text{vector bundles on } X
}}\right\}
&\rightleftharpoons
\left\{{\substack{
  \text{Functions on $n{\hbox{-}}$dimensional} \\
  \text{irreducible Galois reps}
}}\right\}\\
T_x\curvearrowright x &\rightleftharpoons\operatorname{Frob}_x \curvearrowright?
\end{align*}
We regard the LHS \(A\) as ``automorphic forms''.

This suggests the following viewpoint on the Langlands correspondence:
we are only seeing one level, and there is a \emph{second} arithmetic
field theory \(B^{(G {}^{ \vee })}\) built out of Galois representations
of the Langlands dual \(G {}^{ \vee }\), so \({\mathbb{Z}}\) yields a
vector space and \({ {\mathbb{Q}}_p }\) yields a category, and an
equivalence of arithmetic field theories
\(A^{(G)} \rightleftharpoons B^{(G {}^{ \vee })}\). Often \(B\) is a
category of coherent sheaves. This should package local, global, and
geometric Langlands into a single theory!

\end{remark}

\begin{remark}

The abstract correspondence between automorphic forms and Galois reps
isn't so useful; the real utility comes from matching structures and
numerical invariants on both sides, e.g.~Fourier coefficients,
Rankin-Selberg or doubling integrals, the \(\Theta\) correspondence, etc
which all match with something on the Galois side (usually an
\(L{\hbox{-}}\)function). This yields a panoply of matching invariants!
E.g. for \(E/{\mathbb{Q}}\) an elliptic curve,
\begin{align*}
L(\operatorname{Sym}^2 E, 1) = \prod_p {p^2 \over (1-1/p) {\sharp}E({\mathbb{F}}_{p^2})} \in \pi\cdot \mathrm{Area}(E_{\mathbb{C}}) {\mathbb{Q}}
,\end{align*}
where the area is of the fundamental parallelogram of \(E\), which is
hard to prove without automorphic forms. How can we interpret this in
terms of TQFTs?

\end{remark}

\begin{remark}

Consider numerical invariants of automorphic forms and Galois reps
landing in \({\mathbb{C}}\). Let \({\mathcal{O}}\) be a 3-dimensional
ring of integers over \(X\).

The numerical invariants of Galois reps should be elements of
\(B_{\mathcal{O}}^{(G {}^{ \vee })}\), and numerical invariants of
automorphic forms should come from \(A_{\mathcal{O}}^{(G)}\) where given
\(P\), one takes
\(\phi\mapsto {\left\langle {P},~{\phi} \right\rangle}\). To find
matching invariants, we want to match elements in \(A_{\mathcal{O}}\) to
elements in \(B_{\mathcal{O}}\). More ambitiously, we can ask for
matching \emph{boundary conditions} in \(A^{(G)}\) and
\(B^{(G {}^{ \vee })}\).

\end{remark}

\begin{definition}[Boundary conditions, informal definition]

A \textbf{boundary condition} in \({\mathsf{TQFT}}_4\) is a coherent
assignment:

\begin{itemize}
\tightlist
\item
  3-manifolds \(M \leadsto v\in A_M\) a distinguished vector
\item
  2-manifolds \(S \leadsto X_S\) a distinguished object in \(A_S\)
\end{itemize}

\begin{quote}
See Kasputin's 2010 ICM address for a nice overview.
\end{quote}

\end{definition}

\begin{remark}

Joint work with David Ben-Zvi, Sakellaridis, an informal summary:

\begin{itemize}
\tightlist
\item
  A variety \(G{\hbox{-}}\)variety gives a boundary condition for both
  \(A^{(G)}\) and \(B^{(G {}^{ \vee })}\),
\item
  For suitable choice of \(Y\), this recovers familiar invariants of
  automorphic forms mentioned above,
\item
  On the Galois side this recovers \(L{\hbox{-}}\)functions,
\item
  There is a proposed specific class of dual pairs
  \((G, Y) \rightleftharpoons(G {}^{ \vee }, Y {}^{ \vee })\) which give
  matching/dual invariants. E.g. each periodic integral should have a
  dual.
\end{itemize}

\end{remark}

\begin{question}

For the next generation of number theorists: why are there such
similarities between TQFTs and automorphic forms? This is something deep
that we barely understand at all.

\end{question}

\begin{remark}

Extending to 1-dimensional objects: these should be 2-categories which
are categorical reps of a loop group.

\end{remark}

\hypertarget{ellen-eischen-automorphic-forms-on-unitary-groups-talk-1}{%
\section{Ellen Eischen, Automorphic Forms on Unitary Groups, Talk
1}\label{ellen-eischen-automorphic-forms-on-unitary-groups-talk-1}}

\begin{remark}

Overall plan:

\begin{itemize}
\tightlist
\item
  Introduce automorphic forms on unitary groups,
\item
  Techniques to study \emph{algebraic} aspects of
  \(L{\hbox{-}}\)functions,
\item
  Using unitary groups as a convenient setting -- a large enough class
  of groups to be interesting, but confined enough to be tractable.
\end{itemize}

Today:

\begin{itemize}
\tightlist
\item
  Motivations from modular forms,
\item
  Fundamental definitions.
\end{itemize}

If the previous talk was a ``fairy tale'', this will flesh out the
``based on a true story'' part!

\end{remark}

\hypertarget{motivation-from-modular-forms}{%
\subsection{Motivation from modular
forms}\label{motivation-from-modular-forms}}

\begin{example}[?]

Consider \(\zeta(2k)\) for \(k\in {\mathbb{Z}}_{\geq 0}\); known to
Euler as
\begin{align*}
\zeta(2k) = (-1)^k \pi^{2k} {2^{2k-1} \over (2k-1)!} \qty{-{ {B_{2k}} \over 2k} }
= (-1)^k \pi^{2k} {2^{2k-1} \over (2k-1)!} \zeta(1-2k)
,\end{align*}
where \(B_{2k}\) is the \(2k\)th \textbf{Bernoulli number}, whose
exponential generating function is
\begin{align*}
{ze^z\over e^z-1} = \sum_{k\geq 0} B_k {z^k\over k!} \in {\mathbb{Q}}{\left[\left[ z \right]\right]  }
.\end{align*}

Proving rationality of \(\zeta(2k)\) (up to powers \(\pi^n\)) involves
the normalized Eisenstein series
\begin{align*}
G_{2k}(q) = \zeta(1-2k) + 2\sum_{k\geq 1} \sigma_{2k-1}(n) q^n, \quad q\coloneqq e^{2\pi i z}, \quad \sigma_k(n) \coloneqq\sum_{d\divides n} d^k
,\end{align*}
and one can use similar techniques to prove rationality for

\begin{itemize}
\item
  Dedekind zeta functions for \(K\in \mathsf{Field}_{/ {{\mathbb{Q}}}}\)
  totally real:
  \begin{align*}
  \zeta_K(s) = \sum_{{\mathfrak{a}}{~\trianglelefteq~}{\mathcal{O}}_K} {1\over N({\mathfrak{a}})^s}, \qquad N(I) \coloneqq\operatorname{Nm}_{K/{\mathbb{Q}}}(I)
  ,\end{align*}
  where rationality was proved by realizing it as the constant term of a
  Fourier expansion of an Eisenstein series and studying spaces of
  modular forms.
\item
  \(L{\hbox{-}}\)functions \(L(\chi, s)\) for \(\chi\) a \textbf{Hecke
  character} of a totally real field, and their \(p{\hbox{-}}\)adic
  analogs.
\end{itemize}

All of these correspond to \textbf{Artin \(L{\hbox{-}}\)functions}
\(L(s, \rho)\) for \(\rho\) a Galois representation. Dimensions \(n=1\)
and (partially) \(n=2\) are handled class field theory. Can we generally
show special values are algebraic? And if so, what do these values mean?

\end{example}

\begin{remark}

More generally, one can ask about algebraicity or rationality of special
values of \(L{\hbox{-}}\)functions attached to modular forms. Our first
tool for constructing such things will be \textbf{Rankin-Selberg
convolution}. Why care about special values: Kummer used congruences for
\(\zeta\), checking if \(p\divides { \operatorname{cl}} (K)\) is
equivalent to checking if \(p\) divides numerators of Bernoulli numbers,
which can be used to prove special cases of Fermat. Picked up later for
Iwasawa theory, controls behavior of towers of towers of cyclotomic
extensions in \({\mathsf{G_K}{\hbox{-}}\mathsf{Mod}}\).

\end{remark}

\begin{remark}

Conjectures

\begin{itemize}
\tightlist
\item
  Meanings of \(L{\hbox{-}}\)function values, e.g.~Deligne's conjecture
  that they come from \textbf{motives}.
\item
  Langlands: connections between Galois reps \(\rho\) and automorphic
  reps.
\end{itemize}

\end{remark}

\begin{warnings}

It might seem like \(\operatorname{GL}_n\) for \(n\geq 3\) is the next
step, but this turns out to be too general! Even
\({\operatorname{SL}}_n\) in these ranges is difficult. Instead we'll
move to \textbf{unitary groups}, where we'll have Shimura varieties to
work with.

\end{warnings}

\hypertarget{unitary-groups}{%
\subsection{Unitary groups}\label{unitary-groups}}

\begin{remark}

Fix \(K\in {\mathrm{CM}}\mathsf{Field}\), so \(K/K^+/{\mathbb{Q}}\) with
\(K^+/{\mathbb{Q}}\) totally real and \(K/K^+\) quadratic imaginary, and
\(V\in { \mathsf{Vect} }_{/ {K}}\) with a nondegenerate Hermitian
pairing \({\left\langle {{-}},~{{-}} \right\rangle}\), which can be
extended linearly to \(V_R \coloneqq V\otimes_{K^+} R\) for any
\(R\in\mathsf{Alg}_{/ {K^+}}\).

\end{remark}

\begin{definition}[General Unitary Groups]

The \textbf{general unitary group} is the algebraic group
\(G \coloneqq\operatorname{GU}(V, {\left\langle {{-}},~{{-}} \right\rangle})\)
which is defined for each \(R\in \mathsf{Alg}_{/ {K^+}}\) as
\begin{align*}
R \mapsto \left\{{g\in \operatorname{GL}_{K_R}(V_R) {~\mathrel{\Big\vert}~}{\left\langle {gv},~{gw} \right\rangle} = \nu {\left\langle {v},~{w} \right\rangle} \text{ for some } \nu\in R}\right\}
.\end{align*}
The \textbf{unitary group} is the subgroup for which \(\nu = 1\) is
enforced.

\end{definition}

\begin{remark}

If \(R={\mathbb{R}}\), choose an ordered basis for \(B\) to define the
\textbf{signature}
\begin{align*}
{\left\langle {v},~{w} \right\rangle} = vA { {}^{t}{ (w) } }, \qquad A = {
\begin{bmatrix}
  {\one_a} & {0} 
\\
  {0} & {-\one_b}
\end{bmatrix}
},\qquad
\operatorname{sig}(A) \coloneqq(a, b)
.\end{align*}

For the remainder of today, assume \(K^+ = {\mathbb{Q}}\).

\end{remark}

\hypertarget{automorphic-forms-on-unitary-groups-connections-to-modular-forms}{%
\subsection{Automorphic forms on unitary groups, connections to modular
forms}\label{automorphic-forms-on-unitary-groups-connections-to-modular-forms}}

\begin{remark}

On the modular form side:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(f: {\mathfrak{h}}\to {\mathbb{C}}, f(z) = (cz+d)^{-k}f(\gamma z)\),
  holomorphic at cusps, etc
\item

  \begin{align*}
  \phi_f: {\operatorname{SL}}_2({\mathbb{R}})\to {\mathbb{C}}, {\operatorname{SL}}_2({\mathbb{R}})\curvearrowright{\mathfrak{h}}
  .\end{align*}
  transitively fixing \(i\),
  \begin{align*}
  \phi_f(g) = j(g, i)^{-k}f(gi)
  ,\end{align*}
  and
  \begin{align*}
  \phi_f: \dcosetl{\Gamma}{G({\mathbb{R}})} \to {\mathbb{C}}\\
  \phi_f(g(\operatorname{rot}(\theta)) = e^{ki\theta} \phi_f(g) \qquad \operatorname{rot}(\theta) \coloneqq
  \left[\begin{array}{cc}
  \cos \theta & -\sin \theta \\
  \sin \theta & \cos \theta
  \end{array}\right]
  ,\end{align*}
  extend to
  \begin{align*}
  \phi: \dcosetl{\Gamma Z(G)}{G({\mathbb{R}})} \to {\mathbb{C}}
  \end{align*}
  for
  \(G=\operatorname{GL}_2, {\operatorname{SL}}_2, \operatorname{GL}_2^+\),
  etc
\item
  Adelic interpretation:
  \(\operatorname{GL}_2({\mathbb{A}}) = \operatorname{GL}_2({\mathbb{Q}}) \operatorname{GL}_2^+({\mathbb{R}}) \tilde K\)
  where \(\tilde K \coloneqq\prod_p K_p\)is a compact open subgroups of
  \(\operatorname{GL}_2({ {\mathbb{Q}}_p })\) with determinant
  \({ {\mathbb{Z}}_p }^{\times}\) and equal to
  \(\operatorname{GL}_2({ {\mathbb{Q}}_p })\) for all but finitely many
  places.
\end{enumerate}

\begin{itemize}
\tightlist
\item
  Can recover
  \(\Gamma \coloneqq\operatorname{GL}_2({\mathbb{Q}}) \cap(\operatorname{GL}_2^+({\mathbb{R}}) \times {\mathcal{K}})\)
\item
  Match up
  \begin{align*}
  \dcosetl{\Gamma}{\operatorname{GL}_2({\mathbb{R}})} \rightleftharpoons\dcoset{\operatorname{GL}_2({\mathbb{Q}})}{\operatorname{GL}_2({\mathbb{A}})}{{\mathcal{K}}}
    .\end{align*}
\item
  Get functions
  \begin{align*}
  \phi_f: \dcosetl{\operatorname{GL}_2({\mathbb{Q}})}{\operatorname{GL}_2({\mathbb{A}})} \to {\mathbb{C}}
    .\end{align*}
\end{itemize}

On the automorphic side:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Replace \({\mathfrak{h}}\) with
  \(G/{\mathcal{K}}_\infty = {\operatorname{U}}_{n, m}({\mathbb{R}})/ {\operatorname{U}}_n{ {}^{ \scriptscriptstyle\times^{2} } }\),
  a quotient by a compact.
\item
  Writing \(G\coloneqq\operatorname{GU}(n, m)\) for a form of signature
  \((n,m)\), replace with \(\dcosetl{\Gamma Z(G)}{G}\) where
  \(G\supseteq{\mathcal{K}}_\infty \coloneqq U(n){ {}^{ \scriptscriptstyle\times^{2} } }\),
  and analogously
  \(\dcosetl{\Gamma Z(G)}{G({\mathbb{R}})} \to {\mathbb{C}}\).
\item
  For
  \(G({\mathbb{A}}_f) = {\textstyle\coprod}_i G({\mathbb{R}})^{-1}? {\mathcal{K}}\).
\end{enumerate}

\end{remark}

\begin{remark}

An \textbf{automorphic form} on \({\operatorname{U}}_{n, n}\) is a
holomorphic function \(f\in {\mathfrak{h}}_n\to V\) where
\(\rho\curvearrowright V\) is a representation of
\(\operatorname{GL}_n({\mathbb{C}}){ {}^{ \scriptscriptstyle\times^{2} } }\)
where
\begin{align*}
f(z) = \rho(cz+d, { {}^{t}{ (\mkern 1.5mu\overline{\mkern-1.5muc\mkern-1.5mu}\mkern 1.5mu) } } z + \mkern 1.5mu\overline{\mkern-1.5mud\mkern-1.5mu}\mkern 1.5mu)^{-1}f( \gamma z)\qquad \forall \gamma \in \Gamma = {
\begin{bmatrix}
  {a} & {b} 
\\
  {c} & {d}
\end{bmatrix}
}\in {\operatorname{U}}_{n, n}({\mathcal{O}}_K)
\end{align*}
where
\begin{align*}
\gamma z = (az+b)(cz+d)^{-1}, \qquad {\mathfrak{h}}_n \coloneqq\left\{{z\in \operatorname{Mat}_n({\mathbb{C}}) {~\mathrel{\Big\vert}~}i({ {}^{t}{ (\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu) } } - z) > 0}\right\}
.\end{align*}

\end{remark}

\begin{remark}

One thing we haven't mentioned yet: modular forms as sections of line
bundles over modular curves, so moduli of elliptic curves with level
structure, and the generalized setup will be vector bundles over
(unitary) Shimura varieties.

\end{remark}

\hypertarget{ellen-eischen-talk-2}{%
\section{Ellen Eischen, Talk 2}\label{ellen-eischen-talk-2}}

\begin{remark}

Today: more on automorphic forms, and approaches to studying
\(L(s, \pi)\) for \(\pi\) a cuspidal representation of a unitary group.
Note that we've been taking the adelic approach, see Wee Tek's talk for
how to define \(L{\hbox{-}}\)functions in this setting.

\end{remark}

\begin{remark}

Several perspectives on automorphic forms on unitary groups:

\begin{itemize}
\tightlist
\item
  Functions on generalizations of \({\mathfrak{h}}\),
\item
  Functions on \(G({\mathbb{R}})\) and \(G({\mathbb{A}})\) for
  \(G = {\operatorname{U}}_{a, b}\) a unitary group,
\end{itemize}

Today: sections of a vector bundle over certain moduli spaces.

\begin{quote}
See Shimura's first paper on Rankin-Selberg convolutions! See also two
papers by Siegel's student that define the generalizations
\({\mathfrak{h}}_n\).
\end{quote}

\end{remark}

\hypertarget{modular-forms}{%
\subsection{Modular Forms}\label{modular-forms}}

\begin{definition}[Modular form, geometric definition]

Let \({\mathcal{M}}\) be a modular curve (parameterizing curves with
some level structure) and let \(\xi\to {\mathcal{M}}\) be the universal
elliptic curve. Write \(\Omega_{\xi/{\mathcal{M}}}^1\) for the relative
differentials, and define
\begin{align*}
\omega \coloneqq\pi_* \Omega^1_{\xi/{\mathcal{M}}}
.\end{align*}
A \textbf{modular form} is section of a tensor power of \(\omega\),
i.e.~an element of
\(H^0({\mathcal{M}}; \omega{ {}^{ \scriptstyle\otimes_{}^{k} } })\).

\end{definition}

\begin{remark}

We can regard a modular form as a rule
\((E, \omega) \mapsto F(E, \omega)\in {\mathbb{C}}\) that transforms
like
\begin{align*}
F(E, \lambda \omega) = \lambda^{-k} F(E, \omega) \qquad\forall \lambda\in {\mathbb{C}}^{\times}
.\end{align*}
Equivalently, a rule \(\tilde F\) that maps \(E\) to some
\(\omega\in \Omega_{E/{\mathbb{C}}}\), e.g.~
\begin{align*}
\tilde F(E) = F(E, \omega)\omega{ {}^{ \scriptstyle\otimes_{}^{k} }  }
.\end{align*}

Connecting this with last time:
\begin{align*}
(E, \omega) \rightleftharpoons\Lambda_{(E, \omega)} \rightleftharpoons{\mathbb{Z}}+ \tau {\mathbb{Z}}\\
F(E, \omega) \longrightarrow\cdots \longrightarrow f_F(\tau)
,\end{align*}
i.e.~such a rule can be regarded as a function on lattices.

\end{remark}

\begin{remark}

Similarly, automorphic forms arise as global sections of a vector bundle
over a unitary Shimura variety \({\mathcal{M}}\) parameterizing abelian
varieties with

\begin{itemize}
\tightlist
\item
  A polarization,
\item
  An endomorphism, and
\item
  A level structure.
\end{itemize}

One can similarly identify
\begin{align*}
{\mathcal{M}}({\mathbb{C}}) \cong \dcoset{G({\mathbb{Q}})}{G({\mathbb{A}})}{{\mathcal{K}}\cdot {\mathcal{K}}_\infty}
,\end{align*}
which will be a finite disjoint unions of copies of symmetric spaces
(e.g.~\({\mathfrak{h}}_n\)) for \({\operatorname{U}}_{a, b}\).

\end{remark}

\begin{remark}

Write \(\underline{A}\) for an abelian variety with some extra
structure, one can then also view an automorphic form as a function
\begin{align*}
F(\underline{A}, \ell) = \rho(tg)^{-1}F(\underline{A},\ell) \qquad \forall G\in \operatorname{GL}_a \times \operatorname{GL}_b
,\end{align*}
where \(\ell = (\ell_+, \ell_-)\) is an ordered basis for
\(\Omega_{A/{\mathbb{C}}}\) that decomposes according to the signature.

Let \({\mathcal{A}}\xrightarrow{\pi} {\mathcal{M}}\) be the universal
family and define the sheaf
\(\omega \coloneqq\pi_* \Omega_{{\mathcal{A}}/{\mathcal{M}}}\); one can
then build a sheaf of automorphic forms \(\omega^p\) in much the same
way. This reformulates the notion of an automorphic form in terms of
lattices and functions on symmetric spaces like \({\mathfrak{h}}_n\).

\end{remark}

\hypertarget{section}{%
\subsection{?}\label{section}}

\begin{remark}

Goal for today: introduce an approach to studying certain
\(L{\hbox{-}}\)functions using the \emph{doubling method}. Note that the
example of looking at the constant term of an Eisenstein series from
yesterday turns out to be deceptively simple, hence a different approach
today.

\end{remark}

\begin{example}[Motivating example]

Let - \(f(q) = \sum_{k\geq 1} a_kq^k\) be a weight \(k\) cusp form and -
\(g(q) = \sum_{k\geq 0} b_kq^k\) be a weight \(\ell\) modular form,

where
\(a_k, b_k \in { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }\).
The \textbf{Rankin-Selberg product} is
\begin{align*}
D(s,f,g) = \sum_{n\geq 1} a_n b_n n^{-s}
.\end{align*}

Shimura proved that
\begin{align*}
{ D(m,f,g) \over {\left\langle {f},~{f} \right\rangle}_{{\mathrm{Pet}}} } \in \pi^k { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }\qquad \text{for } \ell < k,\, {k+\ell-2\over 2} < m < k
.\end{align*}

This prove relies on the realization
\begin{align*}
D(k-1-r, f, g) = c\pi^k {\left\langle {\tilde f},~{\, g\delta_{ \lambda}^{(r)} E } \right\rangle}_{{\mathrm{Pet}}}
,\end{align*}
where

\begin{itemize}
\tightlist
\item
  \(E\) is an Eisenstein series of weight \(\lambda = k-\ell - 2r\)
\item
  \(\delta\) is a Maass-Shimura differential operator that raises
  weights by \(2r\),
\item
  \(\tilde f\) is \(f\) with conjugate Fourier coefficients, so
  \(\tilde f(q) \coloneqq\sum_{k\geq 0} \mkern 1.5mu\overline{\mkern-1.5mua_k\mkern-1.5mu}\mkern 1.5mu q^k\)
\item
  \({\left\langle {f},~{g} \right\rangle}_{\mathrm{Pet}}\) is the
  Petersson pairing: for \(F\) a fundamental domain
  \begin{align*}
  {\left\langle {f},~{g} \right\rangle}_{\mathrm{Pet}}\coloneqq\int_F \mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu(z) g(z)\, y^{l-2}\,dx\,dy
  .\end{align*}
\end{itemize}

Note that the weight-raising operator doesn't preserve holomorphicity,
which can be bad for algebraicity results, but it turns out that the
result is ``almost holomorphic''.

\end{example}

\hypertarget{proving-algebraicity-a-recipe}{%
\subsection{Proving Algebraicity: A
Recipe}\label{proving-algebraicity-a-recipe}}

\begin{remark}

A general strategy:

\begin{itemize}
\item
  Find a Petersson-style pairing of automorphic forms, e.g.~integrating
  against an Eisenstein series, which looks like an
  \(L{\hbox{-}}\)function:

  \begin{itemize}
  \tightlist
  \item
    Factors into an Euler product
  \item
    Has a functional equation
  \item
    Can be meromorphically continued to \({\mathbb{C}}\)
  \end{itemize}
\item
  Prove appropriate rationality results for \(E\), e.g.~the higher order
  Fourier coefficients are rational, or \(E\) transforms nicely under a
  differential operator. Fourier coefficients are almost always
  important in this step.
\item
  Express a familiar automorphic \(L{\hbox{-}}\)function in terms of
  this pairing.
\end{itemize}

Note that each step is highly nontrivial, and in some contexts, some
steps haven't even been completed yet. The third step often involves
working one place at a time. Even having all three may not be enough,
sometimes the results one gets aren't amenable to algebraic/geometric
study and are instead only good for analytic purposes.

\end{remark}

\hypertarget{doubling}{%
\subsection{Doubling}\label{doubling}}

\begin{remark}

Setup:

\begin{itemize}
\tightlist
\item
  \(K\in \mathsf{Field}_{/ {{\mathbb{Q}}}}\) a quadratic extension
\item
  \(V\in { \mathsf{Vect} }_{/ {K}} ^{\dim = n}\) with a nondegenerate
  Hermitian pairing \({\left\langle {{-}},~{{-}} \right\rangle}_V\)
\item
  \(G = {\operatorname{U}}(V, {\left\langle {{-}},~{{-}} \right\rangle}_V)\)
\item
  \(W = V{ {}^{ \scriptscriptstyle\oplus^{2} } }\) with the induced
  Hermitian pairing
  \begin{align*}
  {\left\langle {(u, v)},~{(u', v')} \right\rangle}_W \coloneqq{\left\langle {u},~{u'} \right\rangle}_V - {\left\langle {v},~{v'} \right\rangle}_V
  .\end{align*}
\item
  A \emph{doubled} group
  \(H \coloneqq{\operatorname{U}}(W, {\left\langle {{-}},~{{-}} \right\rangle}_W)\)
\item
  An embedding
  \begin{align*}
  {\operatorname{U}}(V, {\left\langle {{-}},~{{-}} \right\rangle}) \times
  {\operatorname{U}}(V, -1\cdot {\left\langle {{-}},~{{-}} \right\rangle})
  \hookrightarrow H
  ,\end{align*}
  which in terms of signatures is
  \(\operatorname{sig}(a, b) \times \operatorname{sig}(b, a) \to \operatorname{sig}(a+b, a+b)\)
\end{itemize}

\end{remark}

\begin{remark}

Next time: we'll introduce the doubling integral after pairing with an
Eisenstein series.

\end{remark}

\hypertarget{ellen-eischen-talk-3}{%
\section{Ellen Eischen, Talk 3}\label{ellen-eischen-talk-3}}

\begin{remark}

Some references on doubling:

\begin{itemize}
\tightlist
\item
  PIatestski-Shapiro, Rallis, \(L{\hbox{-}}\)functions for classical
  groups
\item
  Harris, Shimura varieties for unitary groups and the doubling method
\item
  Garrot, Pullbacks of Eisenstein series
\end{itemize}

\end{remark}

\hypertarget{eisenstein-series}{%
\subsection{Eisenstein Series}\label{eisenstein-series}}

\begin{remark}

We'll continue with the previous setup. Let \(P \leq H\) be parabolic
preserving
\(V^\Delta \coloneqq\left\{{(v,v) {~\mathrel{\Big\vert}~}v\in V}\right\}\).
We get a decomposition \(W = V^\Delta \oplus V_\Delta\) where
\(V_\Delta = \left\{{(v, -v) {~\mathrel{\Big\vert}~}v\in V}\right\}\).
Then
\begin{align*}
P = \left\{{ {
\begin{bmatrix}
  {A} & {*} 
\\
  {0} & {{ {}^{t}{ (\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu^{-1}) } }}
\end{bmatrix}
} {~\mathrel{\Big\vert}~}A\in \operatorname{GL}_n}\right\}
.\end{align*}

Given a Hecke character
\begin{align*}
\chi: \dcosetl{K^{\times}}{{\mathbb{A}}_K^{\times}}\to {\mathbb{C}}
,\end{align*}
view this as a character of \(P\) via
\begin{align*}
P\to \operatorname{GL}_n \xrightarrow{\operatorname{det}} {\mathbb{A}}_K^{\times}\to {\mathbb{C}}\\
{
\begin{bmatrix}
  {A} & {*} 
\\
  {0} & {{ {}^{t}{ (\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu^{-1}) } }}
\end{bmatrix}
} \mapsto A\mapsto \operatorname{det}(A) \xrightarrow{\chi} \chi(\operatorname{det}(A))
.\end{align*}

\end{remark}

\begin{definition}[Siegel Eisenstein series]

For \(s\in {\mathbb{C}}\), let
\begin{align*}
f_{S, \chi} = \operatorname{Ind}_{P({\mathbb{A}})}^{H({\mathbb{A}})} (\chi \cdot {\left\lvert {{-}} \right\rvert}^{-s} ) = \left\{{f: H({\mathbb{A}})\to {\mathbb{C}}{~\mathrel{\Big\vert}~}f(ph) = \chi(p) {\left\lvert {p} \right\rvert}^{-s} f(h)}\right\}
.\end{align*}
Define the \textbf{Siegel Eisenstein series} for \(g\in H\) by
\begin{align*}
E_{F_s, \chi}(g) = \sum_{\gamma \in \dcosetl{ P({\mathbb{Q}}) }{ H({\mathbb{Q}}) } } f_{s, \chi}(\gamma g)
.\end{align*}

\end{definition}

\hypertarget{doubling-integral}{%
\subsection{Doubling Integral}\label{doubling-integral}}

\begin{remark}

Setup:

\begin{itemize}
\tightlist
\item
  \(\pi\) a cuspidal automorphic representation of \(G\)
\item
  \(\tilde \pi\) the contragradient (dual) representation of \(\pi\)
\item
  \(\phi\in \pi\)
\item
  \(\tilde\phi\in \tilde\pi\)
\end{itemize}

Define a zeta integral
\begin{align*}
Z(\phi, \tilde\phi, f_{s, \chi}) \coloneqq\int_{ \dcosetl{(G\times G)({\mathbb{Q}})}{(G\times G)({\mathbb{A}})} } 
E_{f_{s, \chi}} (g_1, g_2) \phi(g_1) \tilde \phi(g_2) \chi^{-1}(\operatorname{det}g_2) \,dg _1 \,dg _2
,\end{align*}
which is against an appropriately normalized Haar measure.

\end{remark}

\begin{remark}

Most analytic properties of \(E\) will carry over to \(Z\), e.g.~the
functional equation and meromorphic continuation -- this is a common
theme! In the case of \(G = \operatorname{GU}_1\) or
\(\operatorname{GU}_n\) a definite unitary group, one can express \(Z\)
as a \emph{finite} sum.

\end{remark}

\begin{theorem}[?]

\begin{align*}
Z(\phi, \tilde\phi, f_{s, \chi}) = \int_{G({\mathbb{A}})} f_{s, \chi}((g, 1)) {\left\langle {\pi(g) \phi},~{\tilde\phi} \right\rangle} \,dg , \qquad {\left\langle {\phi},~{\tilde\phi} \right\rangle} = \int_{\dcosetl{G({\mathbb{Q}})}{G({\mathbb{A}})}} \phi(g)\tilde\phi(g) \,dg 
,\end{align*}
where this pairing is \(G{\hbox{-}}\)invariant which is unique up to a
constant multiple.

\end{theorem}

\begin{corollary}[?]

If there is a restricted tensor product representation
\begin{align*}
\pi = \bigotimes^{\operatorname{res}}_v \pi_v, \qquad 
\tilde \pi = \bigotimes^{\operatorname{res}}_v \tilde \pi_v, 
\end{align*}
where
\(\operatorname{Ind}(\chi{\left\lvert {{-}} \right\rvert}^v)= \bigotimes O_v\)
with

\begin{itemize}
\tightlist
\item
  \(\phi = \bigotimes^{\operatorname{res}}_v \phi_v\),
\item
  \(\tilde \phi = \bigotimes^{\operatorname{res}}_v \tilde \phi_v\),
\item
  \(f_{s, \chi} = \bigotimes^{\operatorname{res}}_v f_v\),
\end{itemize}

then there is an Euler product decomposition
\begin{align*}
Z(\phi,\tilde\phi, f_{s,\chi}) = \prod_v Z_v(\phi_v, \tilde\phi_v, f_v)
\end{align*}
where
\begin{align*}
Z_v(\phi_v, \tilde\phi_v, f_v) \coloneqq\int_{G({\mathbb{Q}}_v)} f_v((g, 1)) {\left\langle {\pi_v(g) \phi_v },~{ \tilde\phi_v} \right\rangle} \,dg 
.\end{align*}

\end{corollary}

\begin{proof}[of corollary]

By the uniqueness of the invariant pairing, there must exist a
decomposition
\begin{align*}
{\left\langle {\phi},~{\tilde\phi} \right\rangle} = \prod_v
{\left\langle {\phi_v},~{\tilde\phi_v} \right\rangle} 
.\end{align*}

\end{proof}

\begin{remark}

Shimura computed coefficients in many cases.

\end{remark}

\begin{remark}

Outline proof of the theorem: we'll analyze the orbits of
\(G\times G\curvearrowright X\coloneqq\dcosetl{P}{H}\). Setup:

\begin{itemize}
\tightlist
\item
  Fix \(X\coloneqq\dcosetl{P}{H}\) and write \(\gamma\in X\),
  identifying it with its coset \(H\gamma\).
\item
  For each \(\gamma\in X\), write
  \([G\times G]^\gamma \coloneqq{\operatorname{Stab}}_{G\times G}(\gamma)\).
\item
  Write \([\gamma]\) for the orbit of \(P\gamma\) under the
  \(G\times G\) action.
\end{itemize}

\end{remark}

\begin{remark}

Idea: write \(E\) as a sum and rearrange, then reduce to a known
computation. Reexpress it as
\begin{align*}
E_{f, \chi} (h) 
&= 
\sum_{\scriptscriptstyle [\gamma] \in \dcoset{P({\mathbb{Q}})} {H({\mathbb{Q}})} {(G\times G)({\mathbb{Q}})} }
\qty{ \sum_{ \scriptscriptstyle \dcosetl{(G\times G)({\mathbb{Q}})} {(G\times G)({\mathbb{Q}})} } f_{s, \chi}( \gamma h)} \\
&=
\sum_{\scriptscriptstyle [\gamma] \in \dcoset{P({\mathbb{Q}})} {H({\mathbb{Q}})} {(G\times G)({\mathbb{Q}})} }
\qty{ \sum_{ \scriptscriptstyle \dcosetl{(G\times G)({\mathbb{Q}})} {(G\times G)({\mathbb{Q}})} } 
\int_{ \scriptscriptstyle \dcosetl{(G\times G)({\mathbb{Q}})} {[G\times G]({\mathbb{A}})} } f_{s, \chi}((g, h)) \phi(g) \tilde\phi(h) \chi^{-1}\operatorname{det}(h) \,dg \,dh  }\\
&\coloneqq
\sum_{\scriptscriptstyle [\gamma] \in \dcoset{P({\mathbb{Q}})} {H({\mathbb{Q}})} {(G\times G)({\mathbb{Q}})} } I(\gamma)
,\end{align*}
where
\begin{align*}
I(\gamma) \coloneqq
\int_{\dcosetl{[G\times G]^\gamma ({\mathbb{Q}})} {[G\times G]({\mathbb{A}})} } f_{s, \chi}(\gamma(g, h)) \phi(g)\tilde\phi(g) \chi^{-1}\operatorname{det}(g)\,dg \,dh 
.\end{align*}

We'll have to analysis the \(\gamma = 1\) and \(\gamma\neq 1\) cases
separately, they're quite different:

\begin{itemize}
\tightlist
\item
  For \(\gamma = 1\), \(I(\gamma)\) will be the RHS in the theorem
  statement.
\item
  For \(\gamma\neq 1\), \(I(\gamma) = 0\).
\end{itemize}

Rewrite the stabilizer in a more convenient way:
\begin{align*}
[G\times G]^{\gamma}
&= \left\{{(g,h) \in G\times G{~\mathrel{\Big\vert}~}P\gamma(g, h) = P\gamma}\right\} \\
&= \left\{{(g, h)\in G\times G{~\mathrel{\Big\vert}~}\gamma(g, h)\gamma^{-1}\in P}\right\}
,\end{align*}
so
\begin{align*}
[G\times G]^1 = P \cap(G\times G) = \left\{{(g,g){~\mathrel{\Big\vert}~}g\in G}\right\} \coloneqq G^\Delta
.\end{align*}
Thus
\begin{align*}
f_{s, \chi}(1\cdot(g, h)) 
&= f_{s, \chi}(g, h) \\
&= f_{s, \chi}((h,h) \cdot (h^{-1}g, 1)) \\
&= \chi(\operatorname{det}h) f_{s,\chi}(h^{-1}g, 1)
\end{align*}
and
\begin{align*}
I(1) = 
\int_{\dcosetl{ G^\Delta({\mathbb{Q}}) } {(G\times G)({\mathbb{A}})} } f_{s, \chi}(h^{-1}g, 1) \phi(g)\tilde\phi(g) \,dg \,dh 
.\end{align*}
We have an identification
\begin{align*}
G\times G &\rightleftharpoons G^\Delta \times(G\times 1) \rightleftharpoons G\times G \\
(g,h) &\rightleftharpoons(h,h)\cdot(h^{-1}g, 1) \rightleftharpoons(h,h^{-1}g) \coloneqq(g,g_1)
,\end{align*}
which we can use to write
\begin{align*}
I(1) 
&= \int_{G({\mathbb{A}})} \int_{\dcosetl{G({\mathbb{Q}})} {G({\mathbb{A}})} } f_{s,\chi}(g_1, 1) \pi(g_1) \pi(h) \tilde\phi(h)\,dh \,dg _1 \\
&= \int_{G({\mathbb{A}})} f_{s, \chi}(g_1, 1) {\left\langle {\pi(g_1) \pi},~{\tilde\phi} \right\rangle} \,dg _1
.\end{align*}

All other orbits \([\gamma]\neq [1]\) decompose to products including
terms of the form
\begin{align*}
\int_{\dcosetl{N_i({\mathbb{Q}})} {N_i({\mathbb{A}})} } \phi_i(n\cdot g)\,dn
\end{align*}
for \(i=1,2\), \(\phi_1 = \phi, \phi_2 = \tilde \phi\) and \(N_i\)
unipotent radicals of a parabolic subgroup of \(G\) which is nontrivial
for at least on term -- however, these are cuspidal, so such integrals
vanish (essentially by definition), making the entire thing vanish.

\end{remark}

\begin{remark}

This falls into step 1 of the overall strategy -- we found a pairing. So
the next question is step 2: can we choose
\(f_{s, \chi}, \phi,\tilde\phi\) so that we nice multiples of Langlands
\(L{\hbox{-}}\)functions \(L(s,\pi,\chi)\)? This will rely on reducing
to computations to local integrals that were computed by Godement and
Jacquet for \(\operatorname{GL}_n\).

Next: pulling back automorphic forms to smaller groups, what does this
look like for \(n=1\)?

\end{remark}

\hypertarget{ellen-eischen-talk-4-revisiting-the-doubling-method-for-n1}{%
\section{\texorpdfstring{Ellen Eischen, Talk 4: Revisiting the Doubling
Method for
\(n=1\)}{Ellen Eischen, Talk 4: Revisiting the Doubling Method for n=1}}\label{ellen-eischen-talk-4-revisiting-the-doubling-method-for-n1}}

\hypertarget{reducing-to-finite-sums}{%
\subsection{Reducing to Finite Sums}\label{reducing-to-finite-sums}}

\begin{remark}

Goal: see what happens if we do the doubling method in the following
setup. Let

\begin{itemize}
\item
  \(n=1\)
\item
  \(K\in \mathsf{Field}_{/ {{\mathbb{Q}}}}\) be an imaginary quadratic
  field
\item
  \(V\in { \mathsf{Vect} }^{\dim = 1}_{/ {K}}\)
\item
  \(W = V{ {}^{ \scriptscriptstyle\oplus^{2} } }\)
\item
  \(G\coloneqq{\operatorname{U}}(V, {\left\langle {{-}},~{{-}} \right\rangle}_V) \cong {\operatorname{U}}_1 = \left\{{g\in \operatorname{GL}_2 {~\mathrel{\Big\vert}~}g\mkern 1.5mu\overline{\mkern-1.5mug\mkern-1.5mu}\mkern 1.5mu = 1}\right\}\).
  Note that
  \(G \subseteq \operatorname{GU}(V, {\left\langle {{-}},~{{-}} \right\rangle}_V)\cong \operatorname{GU}_1 \cong \operatorname{GL}_1\)
\item
  \(H\coloneqq{\operatorname{U}}(W, {\left\langle {{-}},~{{-}} \right\rangle}_V) \cong {\operatorname{U}}_{1, 1}\).
  Note that
  \(H \subseteq \operatorname{GU}(W, {\left\langle {{-}},~{{-}} \right\rangle}_V)\cong \operatorname{GU}_{1, 1}\)
\end{itemize}

\end{remark}

\begin{remark}

Spoiler: we'll get an expression for the \(L{\hbox{-}}\)function
\(L(s, \chi)\) for a Hecke character
\(\chi: \dcosetl{K^{\times}}{{\mathbb{A}}_K^{\times}} \to CC^{\times}\)
as a finite sum of values \(E_\chi(A) \chi(A)\) for some elliptic curves
\(A\) with CM by \({\mathcal{O}}_K\), and we'll obtain an algebraicity
result.

\end{remark}

\begin{remark}

Note that
\begin{align*}
\operatorname{GU}_{1, 1} \cong \dcosetr{\operatorname{GL}_2\times \mathop{\mathrm{Res}}_{K/{\mathbb{Q}}} {\mathbb{G}}_m }{{\mathbb{G}}_m}
,\end{align*}
and the associated symmetric space consists of copies of the upper half
plane \({\mathfrak{h}}_1\). The associated modular form is a modular
form, possibly with mild additional conditions on each component.

\end{remark}

\begin{remark}

Reminder of the doubling method: we had an integral
\begin{align*}
Z(s,\chi,\phi,\tilde\phi) = \int_{\dcosetl{G{ {}^{ \scriptscriptstyle\times^{2} }  } ({\mathbb{Q}})} {{\mathbb{G}}{ {}^{ \scriptscriptstyle\times^{2} }  }({\mathbb{A}})} } E_{f_S, \chi}(g, h) \phi(g) \tilde \phi(h) \chi^{-1}(\operatorname{det}h) \, dg\, dh
.\end{align*}

Some properties:

\begin{itemize}
\tightlist
\item
  \(Z\) is an automorphic form on
  \(\operatorname{GU}_1 = \operatorname{GL}_1\), and thus a Hecke
  character.
\item
  If one chooses \(\phi=\chi^{-1}\) so \(\phi^{-1}= \chi\) (plus some
  compatibility conditions), \(Z\) collapses to a finite sum:
  \begin{align*}
  Z(s,\chi,\phi\tilde\phi) = \sum_{\scriptscriptstyle \dcoset{ G{ {}^{ \scriptscriptstyle\times^{2} }  } ({\mathbb{Q}}) } { G{ {}^{ \scriptscriptstyle\times^{2} }  } ({\mathbb{A}})} { {\mathcal{K}}}} E_{S,\chi}(g,h)\chi^{-1}(g)
  .\end{align*}
\end{itemize}

\end{remark}

\begin{remark}

There is a diagram:

\begin{center}
\begin{tikzcd}
{G({\operatorname{U}}_1{ {}^{ \scriptscriptstyle\times^{2} }  })} && {\operatorname{GU}_{1,1}} & {} \\
\\
{{\operatorname{U}}_1{ {}^{ \scriptscriptstyle\times^{2} }  }} && {{\operatorname{U}}_{1,1}}
\arrow[hook, from=3-1, to=3-3]
\arrow[hook, from=1-1, to=1-3]
\arrow[hook, from=3-3, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMiwwLCJcXEdVX3sxLDF9Il0sWzIsMiwiXFxVX3sxLDF9Il0sWzAsMCwiRyhcXFVfMVxcY2FydHBvd2VyezJ9KSJdLFswLDIsIlxcVV8xXFxjYXJ0cG93ZXJ7Mn0iXSxbMywwXSxbMywxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFsyLDAsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEsMCwiIiwxLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=}{Link
to Diagram}
\end{quote}

Moreover there is an embedding:

\begin{center}
\begin{tikzcd}
{\operatorname{GU}(V)\times\operatorname{GU}(V)} && {\operatorname{GU}(W)} \\
\\
& {{\mathsf{G}}({\operatorname{U}}(V) \times {\operatorname{U}}(-V)) = \left\{{(g,h) \in \operatorname{GU}{ {}^{ \scriptscriptstyle\times^{2} }  } {~\mathrel{\Big\vert}~}\nu(g) = \nu(h)}\right\}}
\arrow[hook', from=3-2, to=1-1]
\arrow[hook, from=3-2, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMSwyLCJcXEcoXFxVKFYpIFxcdGltZXMgXFxVKC1WKSkgPSBcXHRzeyhnLGgpIFxcaW4gXFxHVVxcY2FydHBvd2VyezJ9IFxcc3QgXFxudShnKSA9IFxcbnUoaCl9Il0sWzAsMCwiXFxHVShWKVxcdGltZXNcXEdVKFYpIl0sWzIsMCwiXFxHVShXKSJdLFswLDEsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoiYm90dG9tIn19fV0sWzAsMiwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=}{Link
to Diagram}
\end{quote}

These induce embeddings of corresponding Shimura varieties:

\begin{itemize}
\tightlist
\item
  \({\mathcal{M}}_{G({\operatorname{U}}_1{ {}^{ \scriptscriptstyle\times^{2} } })} \to {\mathcal{M}}_{\operatorname{GU}_1{ {}^{ \scriptscriptstyle\times^{2} } }}\)
  which classifies produces \(A_1\times A_2\) 1-dimensional AVs with PEL
  structures, so elliptic curves with CM by \({\mathcal{O}}_K\),
\item
  \({\mathcal{M}}_{G({\operatorname{U}}_1{ {}^{ \scriptscriptstyle\times^{2} } })} \to {\mathcal{M}}_{\operatorname{GU}_{1, 1}}\)
  which classifies \emph{certain} 2-dimensional AVs with PEL structures.
\end{itemize}

\end{remark}

\begin{remark}

Recall that the adelic points of our quotients are
\({\mathbb{C}}{\hbox{-}}\)points of unitary Shimura varieties, and
\({\mathcal{M}}_{\operatorname{GU}_{1,1}}({\mathbb{C}}) = {\textstyle\coprod}\dcosetl{\Gamma_K}{{\mathfrak{h}}_1}\)
where we mod out by some level. Any
\(z\in {\mathfrak{h}}= {\mathfrak{h}}_1\) corresponds to some
\({\mathbb{C}}{ {}^{ \scriptscriptstyle\times^{2} } }/\left\langle{(z\mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu + \mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu, za + b)}\right\rangle\)
where \(a,b\) are in some \({\mathcal{O}}_K\) lattice. Note the
similarity to
\({\mathbb{C}}/\left\langle{{\mathbb{Z}}+ \tau {\mathbb{Z}}}\right\rangle\)
for elliptic curves.

\end{remark}

\begin{remark}

An upshot is that there are three special things in this case:

\begin{itemize}
\tightlist
\item
  Integral is a finite sum,
\item
  There are only characters,
\item
  We're evaluating at special points!
\end{itemize}

\end{remark}

\begin{remark}

Let \(Z(s, \chi) \coloneqq Z(s, \chi, \phi,\tilde\phi)\). We can choose
\(f_{S,\chi}\) such that
\begin{align*}
Z(s,\chi) = c L(s, \chi)
\end{align*}
for \(c\) a scalar, i.e.~they differ by a multiple. This expresses
\(L(s, \chi)\) as a finite sum of values of
\(E(s,\chi) \cdot \chi({-})\) for \(E\) an automorphic form on
\({\operatorname{U}}_{1, 1}\), so a special kind of modular form. There
is a variant of \emph{Damerell's formula}, which expresses
\(L(s, \chi)\) as such a finite sum where \(E\) is an Eisenstein series
in a space of Hilbert modular forms.

\end{remark}

\hypertarget{rationality-properties-for-eisenstein-series}{%
\subsection{Rationality Properties for Eisenstein
Series}\label{rationality-properties-for-eisenstein-series}}

\begin{remark}

We can obtain an Eisenstein series on
\({\mathfrak{h}}= {\mathfrak{h}}_1\) of the form
\begin{align*}
\sum_{(c,d)\in \Lambda} {\chi(d) \over (cz+d)^k (cz+d)^s}
\end{align*}
where \(\Lambda\) is an appropriate \({\mathcal{O}}_K\) lattice, and for
certain characters will converge for \(\Re(s) + k > 2 = 2n\). This will
have rational Fourier coefficients, and is holomorphic for \(s=0\). As
in the case of modular forms, there is a \(q{\hbox{-}}\)expansion (or
more generally in other signatures, a Fourier-Jacobi expansion)
principle:

\begin{slogan}

Automorphic forms on \({\operatorname{U}}_{n,n}\) are determined by
their \(q{\hbox{-}}\)expansions.

\end{slogan}

In particular, if the coefficients of the \(q{\hbox{-}}\)expansion are
contained in \(R\), then \(f\) is in fact defined over \(R\). Kai-Wen
Lan proved a more general version of this principle for
\({\operatorname{U}}_{a,b}\) with any signature, and showed that
algebraic \(q{\hbox{-}}\)expansions and analytic (i.e.~Fourier)
expansions agree.

So things look good for \(s=0\)!

\end{remark}

\begin{question}

What about \(s\neq 0\), i.e.~when the Eisenstein series is not
holomorphic?

\end{question}

\begin{answer}

We use Mass-Shimura differential operators \(\delta^{(r)}_K\) to relate
\(E\) at \(s\neq 0\) to \(E\) at \(s=0\), where here \(\delta\) raises
weights by \(2r\). For \(F\) a modular form defined over
\({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }\),
Shimura proved the following:
\begin{align*}
{ (\delta_k^{(r)} F)(A) \over \Omega^{k+2r} } \in { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }
\end{align*}
for each CM point \(A\). These operators have incarnations in
\({\operatorname{U}}_{n, m}\) and there are analogous algebraicity
results. In fact,
\begin{align*}
E(z, -r, \chi) = c (-4\pi y)^r \delta_k^{(r)} E(z, 0,\chi)
.\end{align*}
where \(c\) is a nice rational factor. Combining these results yields
\begin{align*}
{L(r, \chi) \over \Omega^{k+2r}}\in {\mathbb{Q}}
.\end{align*}

\end{answer}

\begin{remark}

A word about this operator:
\begin{align*}
\delta_k f = {1\over 2\pi i} \qty{{k\over 2iy} + {\frac{\partial }{\partial z}\,} } f = {1\over 2\pi i}y^{-k} {\frac{\partial }{\partial z}\,} (y^k f), \qquad \delta_k^{(r)} = \delta_k \circ \delta_k \circ \cdots \circ \delta_k
.\end{align*}
Katz's idea: reexpress this operator geometrically over a moduli space
of elliptic curves, or more generally AVs, in terms of the
\textbf{Gauss-Manin connection} and the \textbf{Kodaira morphism}, and a
splitting
\begin{align*}
H^1_{\mathrm{dR}} = \omega \oplus H^{0, 1}
\end{align*}
which preserves algebraicity at CM points.

\end{remark}

\hypertarget{wee-teck-gan-automorphic-forms-and-the-theta-correspondence-talk-1}{%
\section{Wee Teck Gan: Automorphic forms and the theta correspondence
(Talk
1)}\label{wee-teck-gan-automorphic-forms-and-the-theta-correspondence-talk-1}}

\begin{remark}

Goal: reformulating the Ramanujan-Petersson conjecture in terms of
representation theory.

\end{remark}

\hypertarget{the-ramanujan-petersson-conjecture}{%
\subsection{The Ramanujan-Petersson
conjecture}\label{the-ramanujan-petersson-conjecture}}

\begin{remark}

Let \(f: {\mathfrak{h}}\to {\mathbb{C}}\) be a holomorphic cusp form of
weight \(k\) and level 1. Suppose \(f\) is an eigenvector for the Hecke
operator \(T_p\), then \(f\) has a Fourier expansion
\begin{align*}
f(z) = \sum_{k\geq 1} a_k(f) q^n, \qquad q\coloneqq e^{2\pi i z}
,\end{align*}
which can be normalized so that \(a_1(f) = 1\). The remaining
coefficients are then the Hecke eigenvalues, so
\begin{align*}
T_p f = a_p(f) f
.\end{align*}

\end{remark}

\begin{conjecture}[Ramanujan-Petersson]

\begin{align*}
{\left\lvert {a_p(f) } \right\rvert} \leq 2p^{k-1\over 2}
.\end{align*}
This was proved by Deligne as a consequence of the Weil conjectures.
There is an analog for \textbf{Maass forms}, which involves the
\textbf{hyperbolic Laplacian}, which similarly bounds Fourier
coefficients.

\end{conjecture}

\begin{remark}

Error terms come from the cusp forms here. There is a bridge that takes
holomorphic modular forms and Maass forms to the world of automorphic
forms.

\end{remark}

\begin{remark}

Setup: let
\(k\in \mathsf{Field}_{/ {{\mathbb{Q}}}} , v \in {\operatorname{Places}}(K)\)
so that \(k_v\in\mathsf{Loc}\mathsf{Field}\). Define the adeles as
\({\mathbb{A}}\coloneqq\prod^{\operatorname{res}}k_v\) which admits a
diagonal embedding \(k\hookrightarrow{\mathbb{A}}\) with
\(\dcosetl{k}{{\mathbb{A}}}\) compact. Let
\(G\in{\mathsf{Alg}}{\mathsf{Grp}}_{/ {k}}\) be reductive,
e.g.~\({\operatorname{SL}}_n, {\operatorname{U}}_n\), we then similarly
have
\begin{align*}
G(k) \hookrightarrow G({\mathbb{A}}) = \prod^{\operatorname{res}}_v G(k_v)
\end{align*}
with \(\left\{{k_v}\right\}\) an open compact subgroup. Write
\([G] = \dcosetl{G(k)}{G({\mathbb{A}})}\), and note the there is a right
action \([G] \curvearrowleft G({\mathbb{A}})\).

\end{remark}

\hypertarget{automorphic-reps}{%
\subsection{Automorphic Reps}\label{automorphic-reps}}

\begin{definition}[Automorphic forms on reductive groups]

An \textbf{automorphic form} on \(G\) is a function
\(f: [G]\to {\mathbb{C}}\) satisfying

\begin{itemize}
\tightlist
\item
  Regularity conditions: e.g.~at worst polynomial growth \(f\sim z^k\),
  smoothness, and derivatives \(f^{(n)}\sim z^k\) for the same exponent.
\item
  Finiteness conditions: \(K{\hbox{-}}\)finiteness for
  \(K = \prod_v k_v\), or more generally
  \(Z({\mathfrak{g}}){\hbox{-}}\)finiteness.
\end{itemize}

Write \({\mathcal{A}}(G)\) for the vector space of automorphic forms on
\(G\). Note that this carries at left \(G({\mathbb{A}})\) action:
\begin{align*}
(g_0 . f)(g) = f(g g_0)
.\end{align*}

\end{definition}

\begin{remark}

The finiteness condition will guarantee that \(f\) will come from the
kernel of a differential operator, e.g.~the CR equations for holomorphy.
Requiring \(K{\hbox{-}}\)finiteness only gives an action on finite
adeles.

\end{remark}

\begin{definition}[Automorphic representation]

An \textbf{automorphic representation} is an irreducible representation
of \({\mathcal{A}}(A)\).

\end{definition}

\hypertarget{cusp-forms}{%
\subsection{Cusp Forms}\label{cusp-forms}}

\begin{definition}[Cusp forms]

A form \(f\in {\mathcal{A}}(G)\) is \textbf{cuspidal} iff for all
parabolic subgroups \(P\) with \(P=MN\), the constant term of \(f\)
along \(N\) is zero, where the constant term is defined as
\begin{align*}
f_N(g) = \int_{[N]} = f(ng) \,dn
.\end{align*}
This yields a subspace of cusp forms
\({\mathcal{A}}_{ \mathrm{cusp} }(G) \leq {\mathcal{A}}(G)\) which is
stable under the \(G({\mathbb{A}})\) action.

\end{definition}

\begin{remark}

One can take a character \(\psi: [N]\to {\mathbb{C}}^{\times}\), then
there is a \((N, \psi){\hbox{-}}\)Fourier coefficient of \(f\):
\begin{align*}
f_{N, \psi}(g) = \int_{[N]} \mkern 1.5mu\overline{\mkern-1.5mu\psi(n)\mkern-1.5mu}\mkern 1.5mu \cdot f(ng) \,dn
.\end{align*}

\end{remark}

\begin{remark}

Uniform moderate growth and being cuspidal imply that
\(f\in {\mathcal{A}}_{ \mathrm{cusp} }(G)\) rapidly decays at
\(\infty\), i.e.~faster than \(1/p\) for any polynomial, so that
\(f\in L^2\):
\begin{align*}
\int_{[G]} {\left\lvert {f} \right\rvert}^2 < \infty
.\end{align*}
So define the Hilbert space of square-integrable automorphic forms
\begin{align*}
{\mathcal{A}}_2(G) \coloneqq\left\{{f\in {\mathcal{A}}(G) {~\mathrel{\Big\vert}~}{\left\lVert {f} \right\rVert}_{L^2}<\infty }\right\}
.\end{align*}
There is a containment
\begin{align*}
{\mathcal{A}}_{ \mathrm{cusp} }(G) \subseteq {\mathcal{A}}_2(G) \subseteq {\mathcal{A}}(G)
,\end{align*}

where there are a decomposition into irreducible reps

\begin{itemize}
\tightlist
\item
  \({\mathcal{A}}_{ \mathrm{cusp} }(G) = \bigoplus_\pi m(\pi) \pi\) for
  some cuspidal multiplicities \(m\),
\item
  \({\mathcal{A}}_2(G) = \bigoplus_\pi m(\pi)\pi\) for some \(L^2\)
  multiplicities \(m\).
\end{itemize}

\end{remark}

\begin{question}

A main question for automorphic representations: for which \(\pi\) is
\(m(\pi) > 0\)? I.e. which representations occur as cuspidal or \(L^2\)
reps? Moreover, what do all of the irreducible reps of
\(G({\mathbb{A}})\) look like?

\end{question}

\begin{remark}

Recall that since
\(G({\mathbb{A}}) = \prod^{\operatorname{res}}_v G(k_v)\), we expect a
representation \(\pi\) of \(G({\mathbb{A}})\) to break up as
\(\pi = \bigotimes^{\operatorname{res}}_v \pi _v\) with

\begin{itemize}
\tightlist
\item
  \(\pi_v\in {\mathsf{Irr}}(G(k_v))\),
\item
  \(\pi_v^{k_v} \neq 0\) for almost all \(v\), so \(k_v\) is
  \textbf{unramified} or \textbf{spherical}.
\end{itemize}

\end{remark}

\hypertarget{unramified-reps}{%
\subsection{Unramified Reps}\label{unramified-reps}}

\begin{remark}

There is a containment \(G_v \supseteq K_v\) where \(G_v\) is
unramified, i.e.~quasi-split (so has a Borel) and split by an unramified
extension of \(k_v\), and \(K_v\) is a \textbf{hyper-special} subgroup,
which is a maximal compact. This yields
\(G_v \supseteq B_v\supseteq T_v N_v\), and there is a bijection
\begin{align*}
{\mathsf{Irr}}{\mathsf{Rep}}(G_v)(K_v{\hbox{-}}\text{unramified}) &\rightleftharpoons
\left\{{\text{Unramified characters of }T_v}\right\}/W \\
I(\chi) = \operatorname{Ind}_{B_v}^{G_v} \chi &\mapsfrom \chi
,\end{align*}
where we mod out by a Weyl group action \(W\). Note that \(I(\chi)\) is
the unique unramified subquotient.

There is a further correspondence
\begin{align*}
\left\{{\text{Unramified characters of }T_v}\right\}/W \\
&\underset{ {\text{Langlands}} }{\rightleftharpoons}
\left\{{\text{Semisimple conjugacy classes in } G {}^{ \vee }({\mathbb{C}})}\right\} \\
\chi &\mapsto S_\chi
,\end{align*}
so there is some semisimple conjugacy class associated to characters
\(\chi\).

\end{remark}

\begin{remark}

Thus for \(\pi\in {\mathcal{A}}_{ \mathrm{cusp} }(G)\) with
\(\pi = \bigotimes^{\operatorname{res}}_v \pi_v\), one gets a collection
\(\left\{{S_{\pi_v} {~\mathrel{\Big\vert}~}v\not\in S}\right\} \subseteq G {}^{ \vee }\).
For \(R: G {}^{ \vee }\to \operatorname{GL}_N({\mathbb{C}})\), we can
form an \(L{\hbox{-}}\)function
\begin{align*}
L^S(s, \pi R) \coloneqq\prod_{v\not\in S} L(s, \pi_v, R),\qquad L(s, \pi_v, R) \coloneqq{1\over \operatorname{det}1-q_v^{-s} R( S_{\pi_v} ) }, \quad q\coloneqq?
.\end{align*}
These generalize Hecke \(L{\hbox{-}}\)functions and those attached to
modular forms.

\end{remark}

\hypertarget{tempered-reps}{%
\subsection{Tempered Reps}\label{tempered-reps}}

\begin{remark}

A character of the torus \(\chi: T_v\to {\mathbb{C}}^{\times}\) yields
\(\pi_\chi\) a \(K_v{\hbox{-}}\)unramified irrep. Say \(\pi_\chi\) is
\textbf{tempered} iff \(\chi\) is unitary, i.e.~it factors as
\(\chi: T_v\to S^1\) so that \({\left\lvert {\chi} \right\rvert} = 1\).
Tempered reps naturally occur as regular representations.

\end{remark}

\begin{remark}

Note that tempered reps are \emph{weakly} contained in \(L^2(G_v)\), but
not e.g.~the trivial representation of \({\operatorname{SL}}_2\) is not
in \(L_2({\mathbb{R}})\), but \({\operatorname{SL}}_2({\mathbb{R}})\)
does not have finite volume. In general, the trivial representation is
not tempered unless the group is compact.

\end{remark}

\begin{conjecture}[Ramanujan-Petersson, reformulated but false]

Let
\(\pi = \bigotimes^{\operatorname{res}}_v \subseteq {\mathcal{A}}_{ \mathrm{cusp} }(G)\)
for \(G\) quasi-split (or split), then \(\pi_v\) is tempered for almost
all \(v\).

\end{conjecture}

\begin{remark}

This conjecture is false! There is a counterexample for
\(G = {\operatorname{SP}}_4\), and a goal for this course is to
construct a counterexample for \(G = {\operatorname{U}}_3\).

\end{remark}

\begin{conjecture}[Ramanujan-Petersson, reformulated and fixed]

If \(\pi \subseteq {\mathcal{A}}_{ \mathrm{cusp} }(G)\) and \(\pi\) is
\textbf{globally generic} (a certain big enough Fourier coefficient),
then \(\pi_v\) is tempered for almost all \(v\).

\end{conjecture}

\hypertarget{unitary-groups-1}{%
\subsection{Unitary groups}\label{unitary-groups-1}}

\begin{definition}[Unitary Groups]

Let

\begin{itemize}
\tightlist
\item
  \(E/F\) be a quadratic extension,
\item
  \({ \mathsf{Gal}} (E/F) = \left\langle{c}\right\rangle\) is cyclic,
\item
  \(V\in { \mathsf{Vect} }_{/ {E}}\),
\item
  \({\left\langle {{-}},~{{-}} \right\rangle}: V{ {}^{ \scriptscriptstyle\times^{2} } }\to E\)
  which is \({\varepsilon}{\hbox{-}}\)Hermitian for
  \({\varepsilon}= \pm 1\), i.e.~
  \begin{align*}
  {\left\langle {av},~{bw} \right\rangle} = a{\left\langle {v},~{w} \right\rangle} b^c, \qquad {\left\langle {v},~{w} \right\rangle} = {\varepsilon}{\left\langle {w},~{v} \right\rangle}
  .\end{align*}
\item

  \begin{align*}
  \delta \in E_0^{\times}\coloneqq\left\{{x\in E^{\times}{~\mathrel{\Big\vert}~}\operatorname{Tr}(x) = 0}\right\}
  .\end{align*}
\item
  \(\delta\cdot{\left\langle {{-}},~{{-}} \right\rangle}\) is
  \((-{\varepsilon}){\hbox{-}}\)Hermitian.
\end{itemize}

Then define the \textbf{unitary group} as
\begin{align*}
{\operatorname{U}}(V) \coloneqq\mathop{\mathrm{Aut}}(V, {\left\langle {{-}},~{{-}} \right\rangle})
.\end{align*}

\end{definition}

\begin{remark}

There are some invariants:

\begin{itemize}
\tightlist
\item
  \(n=\dim V\),
\item

  \begin{align*}
  {\operatorname{disc}}(V) = (-1)^{n\choose 2} \operatorname{det}(V)\in F^{\times}/ \operatorname{Nm}(E^{\times})
  ,\end{align*}
  where the quotient by the image of the norm map is needed to make it
  well-defined.
\end{itemize}

Henceforth we'll take \(V\) to be Hermitian and \(W\) to be
skew-Hermitian.

\end{remark}

\hypertarget{wee-teck-gan-talk-2}{%
\section{Wee Teck Gan (Talk 2)}\label{wee-teck-gan-talk-2}}

\begin{remark}

Correction from last time:
\begin{align*}
{\left\langle {v_2},~{v_1} \right\rangle} = {\varepsilon}{\left\langle {v_1},~{v_2} \right\rangle}^c
.\end{align*}

Notation from last time:

\begin{itemize}
\tightlist
\item
  \(V\) Hermitian, \(W\) skew-Hermitian
\item
  An invariant
  \({\operatorname{disc}}(V) \coloneqq(-1)^m \operatorname{det}V\in F^{\times}/\operatorname{Nm}(E^{\times})\)
  where \(m\coloneqq{n\choose 2}\) and \(n\coloneqq\dim V\)
\item
  \({\operatorname{disc}}(W) = {\operatorname{disc}}( \delta^{-n} V)\)
  where \(\delta\in E_0^{\times}\).
\end{itemize}

\end{remark}

\begin{fact}

Over \(p{\hbox{-}}\)adic fields, \({\operatorname{disc}}(V)\) determines
\(V\). By composing with a quadratic character \(w_{E/F}\), we obtain
\begin{align*}
{\operatorname{disc}}\circ w_{E/F} = (V \xrightarrow{{\operatorname{disc}}} F^{\times}/\operatorname{Nm}(E^{\times})) \xrightarrow{w_{E/F}} \left\langle{\pm 1}\right\rangle
,\end{align*}
so there are exactly two classes of Hermitian vector spaces of a given
dimension, which we'll denote \(V^+, V^-\).

\end{fact}

\begin{remark}

Over a real field, this is not enough -- one also needs the signature
\(\operatorname{sig}(V) = (p, q)\) where \(p+q=n\), in which case
\begin{align*}
{\operatorname{disc}}(V_{p, q}) = (-1)^q (-1)^{p\choose 2}
.\end{align*}
For \(E/K\) an extension of number fields, there is a local-global
principle:
\begin{align*}
{\mathsf{Herm}}{ \mathsf{Vect} }_{/ {K}}  &\hookrightarrow\prod_{v\in {\operatorname{Places}}(K)} {\mathsf{Herm}}{ \mathsf{Vect} }_{/ {K_v}}  \\
V &\mapsto \left\{{V\otimes_K K_v}\right\}_{v\in {\operatorname{Places}}(K)}
.\end{align*}

We'll call spaces in the image of this correspondence \textbf{coherent}.

\end{remark}

\begin{fact}

\(V\) is coherent iff for almost every place \(v\),
\begin{align*}
V\text{ is coherent} \iff 
{\varepsilon}(V_v) = 1 { \text{a.e.} }\text{ and }
\prod_v {\varepsilon}(V_v) = 1
.\end{align*}

\end{fact}

\begin{example}[of classification]

Let \(k\) be a \(p{\hbox{-}}\)adic field.

In rank 1:

\begin{itemize}
\tightlist
\item
  \(E_0^{\times}/ \operatorname{Nm}(E^{\times}) = \left\{{ \delta, \delta'}\right\}\),
\item
  \(W_1^+ = \left\langle{ \delta}\right\rangle\),
\item
  \(W_1^- = \left\langle{ \delta'}\right\rangle\).
\end{itemize}

In rank 2:

\begin{itemize}
\tightlist
\item
  \({\mathbb{H}}= W_2^+ = E e_1 + E e_2\),
\item
  \({\left\langle {e_i},~{e_i} \right\rangle} = 0\) and
  \({\left\langle {e_1},~{e_2} \right\rangle} = 1\), which yields matrix
  \({ \begin{bmatrix}  {0} & {1} \\  {-1} & {0} \end{bmatrix} }\),
\item
  \(W_2^-\) is described by a quaternionic division algebra.
\end{itemize}

In rank \(2n\):

\begin{itemize}
\tightlist
\item
  \(W_{2n}^+ = {\mathbb{H}}{ {}^{ \scriptscriptstyle\oplus^{n} } }\)
\item
  \(W_{2n}^- = W_2^- \oplus {\mathbb{H}}{ {}^{ \scriptscriptstyle\oplus^{(n-1)} } }\)
\end{itemize}

In rank \(2n+1\):

\begin{itemize}
\tightlist
\item
  \(W_{2n+1}^+ = \left\langle{ \delta}\right\rangle \oplus {\mathbb{H}}{ {}^{ \scriptscriptstyle\oplus^{n} } }\),
\item
  \(W_{2n+1}^- = \left\langle{ \delta' }\right\rangle \oplus {\mathbb{H}}{ {}^{ \scriptscriptstyle\oplus^{n} } }\).
\end{itemize}

\end{example}

\hypertarget{howe-ps-counterexample-to-the-ramanujan-petersson-conjecture}{%
\subsection{Howe-PS: Counterexample to the Ramanujan-Petersson
Conjecture}\label{howe-ps-counterexample-to-the-ramanujan-petersson-conjecture}}

\begin{remark}

Let \(\dim W = 3\), so \(U(W) = {\operatorname{U}}_3\), then
\(\mathop{\mathrm{Res}}_{E/K}(W)\in { \mathsf{Vect} }^{\dim = 6}_{/ {k}}\).
The trace to \(K\) yields a symplectic form:
\begin{align*}
\omega({-}, {-}) \coloneqq\operatorname{Tr}_{E/k}{\left\langle {{-}},~{{-}} \right\rangle}_W
.\end{align*}

There is an embedding
\(U(W) \hookrightarrow{\mathsf{Sp}}(\mathop{\mathrm{Res}}_{E/k}(W))\),
so \({\operatorname{U}}_3 \hookrightarrow{\mathsf{Sp}}_6\) There is a
simple something:
\begin{align*}
\Omega \subseteq {\mathcal{A}}_2({\mathsf{Sp}}({-}))
,\end{align*}
which we'll call \textbf{theta functions}. Note that
\(ZU(W) = E^1 \coloneqq\left\{{x\in E^{\times}{~\mathrel{\Big\vert}~}\operatorname{Nm}(x) = 1}\right\}\).
Consider \(i^* \Omega \subseteq {\mathcal{A}}(U(W))\); there is a
central character decomposition
\begin{align*}
i^*( \Omega) = \bigoplus _{\chi} \Omega_{\chi}
,\end{align*}
where the sum is over automorphic characters of \(E^1\).

\begin{claim}

\(\Omega_\chi\) is an irreducible cuspidal representation, with at most
one exception \(\chi\), and this \(\Omega_\chi\) produced a
counterexample for the RP conjecture.

\end{claim}

\end{remark}

\begin{warnings}

A complication: the theta functions don't live on \({\mathsf{Sp}}_6\),
but rather on a double cover, and this leads to many technicalities.

\end{warnings}

\begin{remark}

Howe-PS produces a correspondence:
\begin{align*}
\left\{{\text{Automorphic characters on } E^1\cong {\operatorname{U}}_1}\right\} &\rightleftharpoons\left\{{\text{Automorphic reps of } {\operatorname{U}}_3}\right\} \\
\chi &\mapsto \Omega_{\chi}
.\end{align*}

\end{remark}

\begin{question}

How can one produce an injective map
\begin{align*}
{\mathsf{Irr}}(G) \hookrightarrow{\mathsf{Irr}}(H)
?\end{align*}

\end{question}

\begin{answer}

Recall that
\begin{align*}
{\mathsf{Irr}}(G\times H) = \left\{{\pi \otimes\sigma {~\mathrel{\Big\vert}~}\pi \in {\mathsf{Irr}}(G), \sigma \in {\mathsf{Irr}}(H)}\right\}
.\end{align*}
The idea to produce this map: find \((G\times H){\hbox{-}}\)reps
\(\Omega\) and produce a subset
\begin{align*}
\Sigma_{ \Omega} = \left\{{(\pi, \sigma) {~\mathrel{\Big\vert}~}\mathop{\mathrm{Hom}}_{G\times H}(\Omega, \pi\otimes\sigma) \neq 1}\right\} \subseteq {\mathsf{Irr}}(G) \times {\mathsf{Irr}}(H)
.\end{align*}

\end{answer}

\begin{question}

Is the correspondence \(\Sigma_{\Omega}\) a graph?

\end{question}

\begin{remark}

There is a decomposition
\begin{align*}
\Omega_{G\times H}
&= \bigoplus _\pi \bigoplus _\sigma m(\pi, \sigma) \pi\otimes\sigma \\
&= \bigoplus _\pi \qty{\bigoplus _\sigma m(\pi, \sigma) \sigma }\otimes\pi \\
&\coloneqq\bigoplus _\pi \Theta(\pi) \otimes\pi 
.\end{align*}
Is \(\Theta(\pi)\) an irreducible rep, or zero? If so, this produces a
map
\begin{align*}
\Theta: {\mathsf{Irr}}(G) \to {\mathsf{Irr}}(H)\cup\left\{{0}\right\}
.\end{align*}

\end{remark}

\begin{remark}

Upshot: one needs \(\dim \Omega\) to be small. Suppose
\(G\times H \to E\), take the smallest non-trivial representation
\(\Omega\) of \(E\) and pull it back to \(G\times H\). If
\(G\times H \subseteq E\), this can be done by restriction.

\end{remark}

\begin{remark}

The \textbf{theta correspondence} is an instance of all of these ideas.

\end{remark}

\hypertarget{the-theta-correspondence}{%
\subsection{The Theta Correspondence}\label{the-theta-correspondence}}

\begin{remark}

Let

\begin{itemize}
\tightlist
\item
  \(F\in\mathsf{Field}\) be a \(p{\hbox{-}}\)adic,
\item
  \(E/F\) a quadratic extension,
\item
  \(V\) Hermitian and \(W\) skew-Hermitian so that \(V \otimes_E W\) is
  skew-Hermitian under the symplectic form induced by the trace,
\end{itemize}

This yields a map of the form \(G\times H\to E\):
\begin{align*}
{\operatorname{U}}(V) \times {\operatorname{U}}(W) \to {\mathsf{Sp}}(V\otimes_E W)
.\end{align*}
What is \(\Omega\)? To get small enough weights, one needs to pass to
the \textbf{metaplectic cover} \({\operatorname{Mp}}\).

\end{remark}

\hypertarget{metaplectic-groups-and-weil-reps}{%
\subsection{Metaplectic Groups and Weil
Reps}\label{metaplectic-groups-and-weil-reps}}

\begin{remark}

For \(\psi: F\to {\mathbb{C}}^{\times}\) a nontrivial character:

\begin{center}
\begin{tikzcd}
    {S^1} && {{\operatorname{Mp}}(V\otimes_E W)} \\
    &&&& {\operatorname{GL}(S)} \\
    && {{\mathsf{Sp}}(V\otimes_E W)}
    \arrow["{\Omega = \omega_\psi}", from=1-3, to=2-5]
    \arrow[two heads, from=1-3, to=3-3]
    \arrow[from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJTXjEiXSxbMiwwLCJcXE1wKFZcXHRlbnNvcl9FIFcpIl0sWzIsMiwiXFxTcChWXFx0ZW5zb3JfRSBXKSJdLFs0LDEsIlxcR0woUykiXSxbMSwzLCJcXE9tZWdhID0gXFxvbWVnYV9cXHBzaSJdLFsxLDIsIiIsMix7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFswLDFdXQ==}{Link
to Diagram}
\end{quote}

Here \(\left\{{\omega_\psi}\right\}\) is the smallest
infinite-dimensional representation of \({\operatorname{Mp}}\) and
referred to as the \textbf{Weil representation}.

\end{remark}

\begin{remark}

On where this comes from: QM. One looks at the Heisenberg group, uses
the Stone-von-Neumann theorem, see 2.3 and 2.4 in the notes.

\end{remark}

\begin{remark}

One needs a lift of the following form:

\begin{center}
\begin{tikzcd}
    && {{\operatorname{Mp}}(V\otimes_E W)} \\
    \\
    {{\operatorname{U}}(V)\times {\operatorname{U}}(W)} && {{\mathsf{Sp}}(V\otimes_E W)}
    \arrow["i", from=3-1, to=3-3]
    \arrow["{\exists\, \tilde i?}", from=3-1, to=1-3]
    \arrow[from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMiwwLCJcXE1wKFZcXHRlbnNvcl9FIFcpIl0sWzIsMiwiXFxTcChWXFx0ZW5zb3JfRSBXKSJdLFswLDIsIlxcVShWKVxcdGltZXMgXFxVKFcpIl0sWzIsMSwiaSJdLFsyLDAsIlxcZXhpc3RzXFwsIFxcdGlsZGUgaT8iXSxbMCwxXV0=}{Link
to Diagram}
\end{quote}

By Xudle, \(\tilde i\) exists and is determined by a pair of characters
\((\chi_V, \chi_W)\) of \(E^{\times}\) such that

\begin{itemize}
\tightlist
\item
  \({ \left.{{ \chi_V}} \right|_{{F^{\times}}} } = \omega_{E/F}^{\dim V}\)
\item
  \({ \left.{{\chi_W}} \right|_{{F^{\times}}} } = \omega_{E/F}^{\dim W}\)
\end{itemize}

Such a \(\chi_V\) gives
\(\tilde {\operatorname{U}}(W) \to {\operatorname{Mp}}\), and similarly
for \(W\).

\end{remark}

\begin{remark}

Set
\begin{align*}
\Omega_{V, W, \chi_V, \chi_W, \psi} \coloneqq\tilde\iota_{\chi_V, \chi_W}^*(\omega_\psi)
,\end{align*}
which has properties described in the lecture notes.

\end{remark}

\begin{definition}[The big theta lift as a multiplicity space]

For \(\pi \in {\mathsf{Irr}}{\operatorname{U}}(V)\), define
\begin{align*}
\Theta(\pi) \coloneqq{\operatorname{coinv}}_{{\operatorname{U}}(V)}(\Omega \otimes\pi  {}^{ \vee })
.\end{align*}

\end{definition}

\begin{remark}

Note that there is a \({\operatorname{U}}(W)\) action on both sides.
Moreover,
\begin{align*}
\mathop{\mathrm{Hom}}( {\operatorname{coinv}}_G (\Omega\otimes\pi {}^{ \vee }), {\mathbb{C}}) 
\cong \mathop{\mathrm{Hom}}_G(\Omega\otimes\pi {}^{ \vee }, {\mathbb{C}}) 
\cong \mathop{\mathrm{Hom}}_G(\Omega, ?)
.\end{align*}

\end{remark}

\begin{theorem}[Howe-Kudla]

\envlist

\begin{itemize}
\tightlist
\item
  \(\Theta(\pi)\) has finite length as a \({\operatorname{U}}(W)\) rep,
  and thus has finitely many irreducible quotients.
\item
  For any pair \((\pi, \sigma)\),
  \begin{align*}
  \dim \mathop{\mathrm{Hom}}_{{\operatorname{U}}(V)\times {\operatorname{U}}(W)}(\Omega,\pi\otimes\sigma) < \infty
  .\end{align*}
\end{itemize}

\end{theorem}

\begin{definition}[Small theta lift]

Define \(\theta(\pi)\) to be the maximal semisimple quotient of
\(\Theta(\pi)\). This is a finite length semisimple rep.

\end{definition}

\begin{theorem}[Howe Duality]

\envlist

\begin{itemize}
\tightlist
\item
  \(\theta(\pi)\) is irreducible if \(\Theta(\pi) \neq 0\).
\item
  Uniqueness:
  \(\theta(\pi) \cong \theta(\pi') \implies \pi \cong \pi'\). Thus
  \(\theta: {\mathsf{Irr}}{\operatorname{U}}(V) \to {\mathsf{Irr}}{\operatorname{U}}(W) \setminus\left\{{0}\right\}\)
  is injective on \(\mathop{\mathrm{supp}}\theta\), those reps which are
  not sent to zero.
\end{itemize}

\end{theorem}

\begin{question}

Is \(\theta(\pi)\) zero or not?

\end{question}

\hypertarget{wee-teck-gan-talk-3}{%
\section{Wee Teck Gan (Talk 3)}\label{wee-teck-gan-talk-3}}

\begin{remark}

Last time: we describe the Howe-PS correspondence
\begin{align*}
\text{Automorphic characters of } {\operatorname{U}}_1
&\rightleftharpoons
\text{Automorphic reps of } {\operatorname{U}}_3 \\
\chi &\mapsto \Omega_\chi
.\end{align*}
A correction: it's not true that \(\Omega_\chi\) is cuspidal except for
at most one \(\chi\); instead if can be cuspidal for many \(\chi\). We
defined \(\Omega, \Theta(\pi)\) with a \({\operatorname{U}}(W)\) action,
and Howe duality which took \(\Theta(\pi)\neq 0\) to a unique
irreducible quotient \(\theta(\pi)\). Thus
\(\Theta: {\mathsf{Irr}}{\operatorname{U}}(V)\hookrightarrow{\mathsf{Irr}}{\operatorname{U}}(W) {\textstyle\coprod}\left\{{0}\right\}\)
is injective away from the zero locus.

\end{remark}

\begin{question}

When is \(\Theta(\pi)\neq 0\)?

\end{question}

\begin{remark}

Let \(\dim W\) be odd, and label \(W_r^{\varepsilon}= 2r+1\). We know
all skew-Hermitian spaces of a particular dimension, so we obtain
towers:

\begin{center}
\begin{tikzcd}
    {{\operatorname{U}}W_r^-} &&&& {{\operatorname{U}}W_r^+} \\
    \vdots && {{\operatorname{U}}(V), \pi} && \vdots \\
    {{\operatorname{U}}W_1^-} &&&& {{\operatorname{U}}W_1^+} \\
    {{\operatorname{U}}W_0^-} &&&& {{\operatorname{U}}W_0^+}
    \arrow["{\oplus {\mathbb{H}}}", from=4-1, to=3-1]
    \arrow["{\oplus {\mathbb{H}}}", from=3-1, to=2-1]
    \arrow[from=2-1, to=1-1]
    \arrow["{\oplus {\mathbb{H}}}"', from=4-5, to=3-5]
    \arrow["{\oplus {\mathbb{H}}}"', from=3-5, to=2-5]
    \arrow[from=2-5, to=1-5]
    \arrow["{\theta_1^+}", dashed, from=2-3, to=3-5]
    \arrow["{\theta_r^-}"', dashed, from=2-3, to=1-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Note that \(W_{r+1}^+ = W_r^+ \oplus {\mathbb{H}}\).

\end{remark}

\begin{question}

Which \(\theta_r^{\varepsilon}(\pi)\) are nonzero?

\end{question}

\begin{theorem}[?]

\envlist

\begin{itemize}
\tightlist
\item
  For \(\pi \in {\mathsf{Irr}}{\operatorname{U}}(V)\) and a fixed
  \({\varepsilon}= 1\), there is a smallest
  \(r_0^{\varepsilon}(\pi) \leq \dim V\) such that this is the first
  occurrence of \(\pi\) in the \({\varepsilon}\) tower,
  i.e.~\(\theta_{r_0^{\varepsilon}(\pi)}^{\varepsilon}(\pi) \neq 0\).
\item
  For all \(r>r_0, \Theta^{\varepsilon}(\pi)\neq 0\),
\item
  If \(\pi\) is a supercuspidal rep, then by Kudla,
  \(\Theta_r^{\varepsilon}(\pi)\) is irreducible and is s.c. at the
  first occurrence but not after.
\end{itemize}

\end{theorem}

\begin{remark}

\envlist

\begin{itemize}
\tightlist
\item
  Nonvanishing is reduced to determining \(r_0^+(\pi)\) and
  \(r_0^-(\pi)\).
\item
  If \(r\geq \dim V\), so \(r\) is in the stable range and
  \(\Theta_r^{\varepsilon}(\pi)\neq 0\).
\end{itemize}

Thus reduces checking infinitely many nonzero conditions to just
computing the values of these two numbers. We can reduce this further to
just checking \emph{one} number by the following:

\end{remark}

\begin{theorem}[Conservation relation (B.Y. Sun, C.B. Zhu, Kudla-Rallis)]

\begin{align*}
\dim W_{r_0^+(\pi)}^+ + \dim W_{r_0^-(\pi)} = 2\dim V + 2
.\end{align*}

\end{theorem}

\begin{corollary}[Dichotomy]

If \(\dim W^+ + \dim W^- = 2\dim V\), then for any
\(\pi\in {\mathsf{Irr}}{\operatorname{U}}(V)\), exactly one of
\(\Theta_{W^+}(\pi)\) or \(\Theta_{W^-}(\pi)\) is nonzero.

\end{corollary}

\begin{example}[?]

Take
\({\operatorname{U}}_1\times {\operatorname{U}}_1 = {\operatorname{U}}(V) \times {\operatorname{U}}(W_0)\)
where \({\operatorname{U}}(V) = E^1\), and let
\(\chi\in {\mathsf{Irr}}E^1\). Then
\begin{align*}
\dim W_{r^+(\chi)}^+ + \dim W_{r^-(\chi)}^- = 4
,\end{align*}
These two dimensions are numbers in \(\left\{{1,3}\right\}\), and
exactly one of \(\theta_0^{\pm}(\chi)\) is nonzero, and for \(r>0\) we
have \(\theta_r^{\varepsilon}(\chi) \neq 0\). Which
\(\theta_0^{\varepsilon}(\chi)\) are nonzero?

\end{example}

\begin{theorem}[Moen, Rogawski?, Hams-Kudla-Sweat]

\begin{align*}
\theta_{V, W_0, \psi}(\pi) \neq 0 \iff {\varepsilon}(v) {\varepsilon}(W_0) = {\varepsilon}_E\qty{ {1\over 2}, \chi_E \chi_W^{-1}, \psi(\operatorname{Trace}_{E/F} (\delta-1)) }
\end{align*}
where \({\varepsilon}_E\) is the local epsilon factor defined in Tate's
thesis. Here \(\chi_E\) is the composite character
\(\chi_E(x) = \chi\qty{x\over x^?}\) defined by

\begin{center}
\begin{tikzcd}
    {E^{\times}/F^{\times}} && {E^1} && {\mathbb{C}}^{\times}\\
    x && {{x\over x^c}}
    \arrow["\chi", from=1-3, to=1-5]
    \arrow["\cong", from=1-1, to=1-3]
    \arrow["{\chi_E}"', curve={height=-30pt}, dashed, from=1-1, to=1-5]
    \arrow[maps to, from=2-1, to=2-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJFXFx1bml0cy9GXFx1bml0cyJdLFsyLDAsIkVeMSJdLFs0LDAsIlxcQ0NcXHVuaXRzIl0sWzAsMSwieCJdLFsyLDEsInt4XFxvdmVyIHheY30iXSxbMSwyLCJcXGNoaSJdLFswLDEsIlxcY29uZyJdLFswLDIsIlxcY2hpX0UiLDIseyJjdXJ2ZSI6LTUsInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFszLDQsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=}{Link
to Diagram}
\end{quote}

The \(\delta\in E_0^{\times}\) appears because a Hermitian space depends
on a choice of a traceless element.

\end{theorem}

\begin{example}[?]

Applying Howe-PS to \({\operatorname{U}}_1\times {\operatorname{U}}_3\):
let \(V = \left\langle{1}\right\rangle = V_0^+\) and
\(\chi \in {\mathsf{Irr}}E^1 = {\mathsf{Irr}}{\operatorname{U}}(V)\).
Since \(\dim W^{\varepsilon}= 3\), \(\Omega^{\varepsilon}\) is
semisimple and decomposes as
\begin{align*}
\Omega^{\varepsilon}= \bigoplus _{\chi\in{\mathsf{Irr}}E^1} \chi \otimes\Theta^{\varepsilon}(\chi)
.\end{align*}

\begin{itemize}
\tightlist
\item
  Since \(\dim V = 1\), we're in the stable range and thus
  \(\Theta^{\varepsilon}(\chi)\neq 0\) for all \(\chi\).
\item
  \(\Theta^{\varepsilon}(\chi)\) is irreducible by Howe duality and s.c.
\item
  If \({\varepsilon}= {\varepsilon}_E\qty{{1\over 2}, \cdots}\) as in
  the theorem, \(\Theta^{\varepsilon}(\chi)\) is non-supercuspidal and
  \(\Theta^{-{\varepsilon}}(\chi)\) is supercuspidal.
\end{itemize}

In fact,
\(\Theta^{\varepsilon}(\chi) \hookrightarrow\operatorname{Ind}_B^{{\operatorname{U}}(W)}\qty{ \chi_v {\left\lvert {{-}} \right\rvert}^{- {1\over 2} } \otimes\chi }\)
where \(B = \operatorname{diag}(a, b, (a^c)^{-1}) + N^+\) (upper
triangular) with \(a\in E^{\times}\) and \(b\in E^1\).

\end{example}

\hypertarget{global-setting}{%
\subsection{Global Setting}\label{global-setting}}

\begin{remark}

For \(K\in \mathsf{Field}_{/ {{\mathbb{Q}}}}\), writing
\(\theta = \prod_v \theta_v\), one might hope for a map
\({\mathsf{Irr}}{\operatorname{U}}(V)({\mathbb{A}}) \to {\operatorname{U}}(W)({\mathbb{A}})\).
Instead, we'll want a map
\begin{align*}
\theta:
\left\{{\text{Automorphic reps of } {\operatorname{U}}(V)}\right\} 
&\to 
\left\{{\text{Automorphic reps of } {\operatorname{U}}(W)}\right\} 
,\end{align*}
i.e.~a concrete way to transfer functions from a space \(X\) to a space
\(Y\). If \(K\in C(X\times Y)\), we can define
\begin{align*}
T_K: C(X) &\to C(Y) \\
T_k(f)(y) &\coloneqq\int_X K(x, y) f(x) \,dx
,\end{align*}
so \(K\) acts like a matrix. In our case, we'll want a lift

\begin{center}
\begin{tikzcd}
    &&& {{\operatorname{Mp}}(V\otimes W)({\mathbb{A}})} \\
    \\
    {({\operatorname{U}}(V)\times {\operatorname{U}}(W))({\mathbb{A}})} &&& {{\mathsf{Sp}}(V\otimes W)({\mathbb{A}})}
    \arrow["\iota", from=3-1, to=3-4]
    \arrow["{\exists \tilde\iota\,?}"', from=3-1, to=1-4]
    \arrow[from=1-4, to=3-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwyLCIoXFxVKFYpXFx0aW1lcyBcXFUoVykpKFxcQUEpIl0sWzMsMCwiXFxNcChWXFx0ZW5zb3IgVykoXFxBQSkiXSxbMywyLCJcXFNwKFZcXHRlbnNvciBXKShcXEFBKSJdLFswLDIsIlxcaW90YSJdLFswLDEsIlxcZXhpc3RzIFxcdGlsZGVcXGlvdGFcXCw/IiwyXSxbMSwyXV0=}{Link
to Diagram}
\end{quote}

Here \(\Omega = \tilde{\iota}^* W_\psi\). For
\(\pi \in {\mathcal{A}}_{ \mathrm{cusp} }({\operatorname{U}}(V))\), we
have a map

\begin{center}
\begin{tikzcd}
    {W_\psi} &&&& {{\mathcal{A}}_2({\operatorname{Mp}}(\cdots))} \\
    \\
    &&&& {C([{\operatorname{U}}(V)\times {\operatorname{U}}(W)])}
    \arrow["{\tilde\iota^*}", from=1-5, to=3-5]
    \arrow[from=1-1, to=1-5]
    \arrow["\theta"', from=1-1, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJXX1xccHNpIl0sWzQsMCwiXFxtY2FfMihcXE1wKCkpIl0sWzQsMiwiQyhbXFxVKFYpXFx0aW1lcyBcXFUoVyldKSJdLFsxLDIsIlxcdGlsZGVcXGlvdGFeKiJdLFswLDFdLFswLDIsIlxcdGhldGEiLDJdXQ==}{Link
to Diagram}
\end{quote}

This yields
\begin{align*}
w_\psi \otimes\pi &\to {\mathcal{A}}({\operatorname{U}}(V)) \\
\phi \otimes f &\mapsto \theta(\phi, f), \qquad \theta(\phi, f)(g) \coloneqq\int_{[{\operatorname{U}}(V)]} \theta(\phi)(g, h) \mkern 1.5mu\overline{\mkern-1.5muf(h)\mkern-1.5mu}\mkern 1.5mu\, dh
.\end{align*}
So define the \textbf{global theta lift of \(\pi\)} as
\begin{align*}
\Theta(\pi) \coloneqq\left\langle{\theta(\phi, f) {~\mathrel{\Big\vert}~}\phi\in w_\phi, f\in \pi}\right\rangle \subseteq {\mathcal{A}}({\operatorname{U}}(W))
.\end{align*}

\end{remark}

\begin{question}

\envlist

\begin{itemize}
\tightlist
\item
  Is \(\Theta(\pi)\) nonzero?
\item
  Does it land in \({\mathcal{A}}_2\) or
  \({\mathcal{A}}_{ \mathrm{cusp} }\)?
\item
  What is the relation with the local picture?
\end{itemize}

\end{question}

\begin{proposition}[?]

If \(\Theta(\pi) \subset {\mathcal{A}}_2({\operatorname{U}}(W))\) is a
proper subset, then \(\Theta(\pi)\) is either zero or isomorphic to
\(\bigotimes_v \theta(\pi_v)\).

\end{proposition}

\begin{theorem}[?]

Let
\(\pi \subseteq {\mathcal{A}}_{ \mathrm{cusp} }({\operatorname{U}}(V))\),

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  There exists a smallest \(r_0 = r_0^{\varepsilon}(\pi)\) such that
  \(\Theta_{r_0}^{\varepsilon}(\pi) \neq 0\). In this case,
  \(\Theta_{r_0}^{\varepsilon}(\pi) \subseteq {\mathcal{A}}_{ \mathrm{cusp} }({\operatorname{U}}(W))\).
\item
  For all \(r > r_0\),\(\Theta_r^{\varepsilon}(\pi)\neq 0\) and is
  noncuspidal, i.e.~not contained in
  \({\mathcal{A}}_{ \mathrm{cusp} }({\operatorname{U}}(W))\).
\item
  For all \(r\geq \dim V\) in the stable range,
  \(0\neq \Theta_r^{\varepsilon}(\pi) \subseteq {\mathcal{A}}_2({\operatorname{U}}(W))\).
  Note that being nonzero follows from 1 and 2.
\end{enumerate}

\end{theorem}

\hypertarget{wee-teck-gan-talk-4}{%
\section{Wee Teck Gan (Talk 4)}\label{wee-teck-gan-talk-4}}

\begin{remark}

Take \(V = \left\langle{1}\right\rangle\) and
\(W_r^{\varepsilon}= W_0^{\varepsilon}\oplus {\mathbb{H}}^r\) which has
dimension \(2r+1\), and let
\(\chi \in {\mathcal{A}}({\operatorname{U}}(V))\). We know
\(\Theta_r^{\varepsilon}(\chi)\neq 0\) for all \(r>0\), which is the
stable range. Note that
\(\Theta_r^{\varepsilon}(\chi) \subseteq {\mathcal{A}}_2( {\operatorname{U}}(W_r^{\varepsilon}))\),
i.e.~these are square-integrable. What happens when \(r=0\)?

\end{remark}

\begin{theorem}[?]

\(\Theta_{W_0^{\varepsilon}}(\chi) \neq 0 \iff\) several conditions
hold:

\begin{itemize}
\tightlist
\item
  For all \(v\), \(\Theta_{W_0}^{{\varepsilon}_v}(\chi_v)\neq 0\), so it
  is controlled by local conditions,
\item
  \(L(1/2, \chi_E \chi_W^{-1})\neq 0\), a global condition.
\end{itemize}

\end{theorem}

\begin{remark}

Note that
\begin{align*}
{\varepsilon}_v &= {\varepsilon}(1/2, \chi_E \chi_{W, v}^{-1}, \phi(\operatorname{Trace}?)) \\
1 = \prod_v {\varepsilon}_v &= {\varepsilon}(1/2, \chi_E \chi_W^{-1})
.\end{align*}

\end{remark}

\begin{proof}[of theorem, sketch]

For \(\psi\in W_\phi\), we produce \(\Theta(\phi)\) and obtain an
integral
\begin{align*}
\Theta(\phi, \chi)(g) = \int_{[{\operatorname{U}}(V)]} \phi(g,h) \chi(h)^{-1}\, dh \in {\mathcal{A}}({\operatorname{U}}(W_0))
.\end{align*}
Is this function nonzero for some \(\phi\)? There isn't a good notion of
Fourier expansion here, so one instead computes
\({\left\langle {\Theta(\phi, \chi)},~{\Theta(\phi, \chi)} \right\rangle}\).
Write \(V^\square = V \oplus -V\), where \(-V\) is \(V\) with the form
negated.

\begin{center}
\begin{tikzcd}
    & {{\operatorname{U}}(V^\square)} && {{\operatorname{U}}(W_0)\times {\operatorname{U}}(W_0)} & \textcolor{rgb,255:red,92;green,92;blue,214}{\Theta(\phi,\chi) \overline{\Theta(\phi,\chi)}} \\
    \\
    \textcolor{rgb,255:red,92;green,92;blue,214}{(\chi, \chi^{-1})} & {{\operatorname{U}}(V)\times {\operatorname{U}}(V^-)} && {{\operatorname{U}}(W_0)^\Delta} \\
    & {}
    \arrow[hook, from=3-4, to=1-4]
    \arrow[hook, from=3-2, to=1-2]
    \arrow[hook, from=3-2, to=1-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

One can them map
\({\operatorname{U}}(V^\square) \to {\operatorname{U}}(W_0)^\square\);
this diagram is called the \textbf{doubling see-saw}. Combining this
with Siegel-Weil associates to the above inner product the doubling zeta
integral \(Z(0, \phi, \chi)\). By Ellen's lectures, this reduces to
computing the central value of an \(L{\hbox{-}}\)function,
\(cL(1/2, \chi_E \chi_W^{-1})\), up to a fudge factor \(c\). The process
is the \textbf{Rallis inner product formula}:
\begin{align*}
{\left\langle {\Theta(\phi,\chi)},~{\Theta(\phi,\chi)} \right\rangle} \leadsto Z(0,\phi,\chi)\leadsto L(1/2, \chi_E \chi_W^{-1})
.\end{align*}

\end{proof}

\hypertarget{howe-ps}{%
\subsection{Howe-PS}\label{howe-ps}}

\begin{remark}

Setup:

\begin{itemize}
\tightlist
\item
  \(V = \left\langle{1}\right\rangle\) a 1-dim space
\item
  \(W = W_0 \oplus {\mathbb{H}}\)
\item
  A nonzero irreducible theta lift
  \(\Theta_W(1) \subseteq {\mathcal{A}}_2({\operatorname{U}}(W))\)
\end{itemize}

We know that the local components are contained in non-tempered
principal series, i.e.~
\begin{align*}
\Theta_W(1)_v \hookrightarrow\operatorname{Ind}_?^{{\operatorname{U}}(W_v)} {\left\lvert {{-}} \right\rvert}_v^+ \otimes 1_v?
.\end{align*}
It only remains to check that happens when this is not cuspidal. If it
is not, then \(\Theta_{W_0}(1) \neq 0\), so pick 2 places \(v_1, v_2\)
of \(K\) and swap the signs on \(W_{0, v_i}\) to produce \(W_0'\), and
run the above argument on \(W' = W_0' \oplus {\mathbb{H}}\).

\end{remark}

\hypertarget{arthurs-conjecture}{%
\subsection{Arthur's Conjecture}\label{arthurs-conjecture}}

\begin{remark}

Goal: classify constituents of \({\mathcal{A}}_2(G)\), i.e.~describe
this as a \(G({\mathbb{A}}){\hbox{-}}\)module. We'll make a basic
hypothesis (global Langlands for \(\operatorname{GL}_n\)) that there
exists a group \(L_F\) (thought of as
\({ \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu/F)\))
such that there is a bijection
\begin{align*}
{\mathsf{Irr}}{\mathsf{Rep}}^{\dim = n} L_F \rightleftharpoons{\mathsf{Rep}}_{ \mathrm{cusp} }\operatorname{GL}_n
,\end{align*}
where for all \(v\) there is a Weil-Deligne group
\(L_{F_v} \approx { \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu_v/F_v)\)
with a map \(L_{F_v} \hookrightarrow L_F\).

\end{remark}

\begin{definition}[Near equivalence]

Two adelic representations \(\pi = \bigotimes_v \pi_v\) and
\(\pi' = \bigotimes_v \pi_v'\) are \textbf{nearly equivalent} iff
\(\pi_v \cong \pi_v'\) for almost all places \(v\).

\end{definition}

\begin{remark}

We can decompose into near-equivalence classes
\({\mathcal{A}}_w(G) = \bigoplus_\psi {\mathcal{A}}_{\psi}\), where
\(\psi: L_F\times {\operatorname{SL}}_2 \to {}^{L}{G}\) is a map to the
Langlands \(L\) group
\({}^{L}{G} = G {}^{ \vee }\rtimes{ \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu/F)\),
such that

\begin{itemize}
\tightlist
\item
  A tempered condition: the image is big, \(\psi(L_F)\) is bounded, and
\item
  Centralizers are small:
  \(Z_{G {}^{ \vee }}/Z_{G {}^{ \vee }}^{\Gamma_F}\) is finite.
\end{itemize}

This has something to do with elliptic \(A{\hbox{-}}\)parameters.

\end{remark}

\begin{question}

Given \(\psi\), how can we describe \({\mathcal{A}}_\psi\)?

\end{question}

\begin{remark}

From \(\psi\) we'll obtain

\begin{itemize}
\tightlist
\item
  a global component group/centralizer
  \(S_\psi = Z_{G {}^{ \vee }}/Z_{G {}^{ \vee }}^{\Gamma_F}\),
\item
  local factors
  \(\psi_v: L_{F_v}\times {\operatorname{SL}}_2 \hookrightarrow L_f \times {\operatorname{SL}}_2 \xrightarrow{\psi} {}^{L}{G}\),
\item
  Local component groups
  \(\pi_0\qty{Z_{G {}^{ \vee }}(\psi_v) / Z(G {}^{ \vee })^{\Gamma_F} }\)
  which are finite?
\item
  \(S_\psi \xrightarrow{\Delta} \prod_v S_{\psi_v} \coloneqq S_{\psi/\Delta}\)
  which is compact
\item
  Quadratic characters
  \({\varepsilon}_\psi: S_\psi \to \left\langle{\pm 1}\right\rangle\).
\end{itemize}

\end{remark}

\begin{remark}

For all \(v\), we should have a finite set of unitary reps of
\(G(F_v)\),
\begin{align*}
\prod_{\psi_v} = \left\{{\pi_{\eta_v} {~\mathrel{\Big\vert}~}\eta_v \in {\mathsf{Irr}}S_{\psi_v}}\right\}
,\end{align*}
i.e.~for almost all \(v\), \(\pi_{1_v}\) is irreducible unramified with
Satake parameters
\begin{align*}
\psi_v\qty{\operatorname{Frob}_v, \operatorname{diag}(q_v^{1\over 2} , q_v^{-{1\over 2} })} \in  {}^{L}{G} 
.\end{align*}

\end{remark}

\begin{observation}

The key point: if \(\psi({\operatorname{SL}}_2) = 1\), then
\(\pi_{1_v}\) is tempered. If not, \(\psi_{1_v}\) is non-tempered.

\begin{quote}
This explains how rigidity obstructs the Ramanujan-Petersson conjecture?
\end{quote}

\end{observation}

\begin{remark}

Set \(\pi_\psi = \bigotimes_v \pi_{\psi_v}\) and let
\begin{align*}
{\mathcal{A}}_\psi = \bigoplus_{\eta \in {\mathsf{Irr}}S_{\psi, ?}} m_\eta \pi_\eta, \qquad m_\eta = \dim \mathop{\mathrm{Hom}}_{S_\psi}({\varepsilon}_\psi, \eta)
.\end{align*}
To define \({\varepsilon}_\psi\), define a map
\begin{align*}
(L_F\times \Omega_2)\times S_{\psi} \xrightarrow{\psi\times \operatorname{id}}  {}^{L}{G} / Z(G {}^{ \vee })^{\Gamma_F} \curvearrowleft{ \operatorname{Ad} }{\mathfrak{g}} {}^{ \vee }
\end{align*}
where
\({\mathfrak{g}} {}^{ \vee }= \mathsf{Lie}(G {}^{ \vee }) = \bigoplus _{i\in I} \rho_i \otimes S_{r_i} \otimes\eta_i\)
for some index set \(I\), and \(S_r\) are \(r{\hbox{-}}\)dimensional
irreps of \({\operatorname{SL}}_2\). Set \(T \subseteq I\) to be the
indices such that \(r_i\) is even, \(\eta_i\) is orthogonal, and
\(\rho_i\) is symplectic, and \({\varepsilon}(1/2, \phi_i) = -1\). Then
define
\begin{align*}
{\varepsilon}_\psi: S_\psi &\to \left\langle{\pm 1}\right\rangle \\
s &\mapsto \prod_{i\in T} \eta_i(s)
.\end{align*}

\end{remark}

\begin{example}[?]

For \(\psi({\operatorname{SL}}_2) = \left\{{1}\right\}\),
\({\varepsilon}_\psi = 1\) and \(T = \emptyset\).

\end{example}

\hypertarget{specializing-to-operatornameu_n}{%
\subsection{\texorpdfstring{Specializing to
\({\operatorname{U}}_n\)}{Specializing to \{\textbackslash operatorname\{U\}\}\_n}}\label{specializing-to-operatornameu_n}}

\begin{remark}

Fix \(G={\operatorname{U}}_n\), let \(E/F\) be an extension, and
\begin{align*}
G {}^{ \vee }= \operatorname{GL}_n({\mathbb{C}}) {~\trianglelefteq~} {}^{L}{G}  = \operatorname{GL}_n({\mathbb{C}}) \rtimes{ \mathsf{Gal}} (E/F)
.\end{align*}
An \(L{\hbox{-}}\)parameter is a map
\begin{align*}
\psi: L_F \times {\operatorname{SL}}_2 \to  {}^{L}{G} 
\end{align*}
where the subset \(L_E \times {\operatorname{SL}}_2\) maps to
\(G {}^{ \vee }\). By an email comment of Benedict Gross, \(\psi\) is
determined by this restriction. Not every such map extends, but
conjugate self-dual reps of sign \((-1)^{n-1}\) will.

\end{remark}

\begin{example}[?]

For \({\operatorname{U}}_3\),
\({ \left.{{\psi}} \right|_{{L_E}} }: L_E\times {\operatorname{SL}}_2 \to \operatorname{GL}_3({\mathbb{C}})\)
which decomposes as
\({ \left.{{\psi}} \right|_{{L_E}} } = \mu \oplus \chi \otimes S_2\)
such that \(\chi\) are not characters of \(L_E\), but rather automorphic
characters of \(\dcosetl{E^{\times}}{{\mathbb{A}}_E^{\times}}\) with
\({ \left.{{\mu}} \right|_{{{\mathbb{A}}_F^{\times}}} } = 1\) and
\({ \left.{{\chi}} \right|_{{{\mathbb{A}}_F^{\times}}} } = \omega_{E/F}\).
For \(S_\psi = \mu_2 \xrightarrow{\Delta} \prod_v S_{\psi_v}\), we have
\(S_{\psi_v} = \mu_2\) if \(v\) is inert in \(E\) and 1 otherwise. Then
\({\varepsilon}_{\psi}: \mu_2 \to \left\langle{\pm 1}\right\rangle\)
which is trivial when \({\varepsilon}(1/2, \chi\mu^{-1}) = 1\) and
nontrivial if this is \(-1\). So
\(\prod_{\psi_v} = \left\{{\pi^+_v, \pi_v^-}\right\}\) if \(v\) is
inert, and just \(\left\{{\pi_v^+}\right\}\) otherwise, meaning
\begin{align*}
m(\pi^{{\varepsilon}})
=
\begin{cases}
1 & \prod_v {\varepsilon}_v = {\varepsilon}(1/2) 
\\
0 & \text{otherwise}.
\end{cases}
.\end{align*}
For almost every \(v\), \(\pi_{1_v} = \pi_v^+\). Something about
\(\operatorname{Ind}_{B_v}^{U_?} \chi {\left\lvert {{-}} \right\rvert}_v^{-{1\over 2}} \otimes\tilde \mu\).
Something about Howe-PS.

\end{example}

\hypertarget{aaron-pollack-modular-forms-on-exceptional-groups-lecture-1}{%
\section{Aaron Pollack: Modular forms on exceptional groups (Lecture
1)}\label{aaron-pollack-modular-forms-on-exceptional-groups-lecture-1}}

\begin{remark}

Plans for lectures:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  What is \({\mathsf{G}}_2\), what are modular forms on it?
\item
  Fourier expansions of modular forms on \({\mathsf{G}}_2\).
\item
  Examples and theorems about modular forms on \({\mathsf{G}}_2\).
\item
  Beyond \({\mathsf{G}}_2\), possibly \({\mathbf{E}}_8\).
\end{enumerate}

\end{remark}

\begin{remark}

First generalize modular forms to modular functions: let
\(f:{\mathfrak{h}}\to {\mathbb{C}}\) be a modular form of level
\(\Gamma\) and weight \(\ell>0\). Define
\begin{align*}
\phi_f: {\operatorname{SL}}_2({\mathbb{R}}) &\to {\mathbb{C}}\\
\phi_f(g) &\coloneqq j(g, z)^{-\ell} f(gz) \\ \\
j\qty{ g = {
\begin{bmatrix}
  {a} & {b} 
\\
  {c} & {d}
\end{bmatrix}
},z} &\coloneqq cz+d
.\end{align*}

Some properties:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Growth: \(\phi_f\) is of moderate growth.
\item
  Invariance: \(\phi_f(\gamma g) = \phi_f(g)\) for all
  \(\gamma\in \Gamma \leq {\operatorname{SL}}_2({\mathbb{Z}})\).
\item
  Equivariance on a compact:
  \(k_\theta \coloneqq{ \begin{bmatrix}  {\cos \theta} & {-\sin \theta} \\  {\sin \theta} & {\cos \theta} \end{bmatrix} }\in {\operatorname{SO}}_2({\mathbb{R}})\)
  satisfies \(\phi_f(g k_\theta) = e^{-i\ell \theta} \phi_f(g)\)
\item
  Operator equation: \(D_{{\mathrm{CR}}} \phi_f \equiv 0\) where we
  decompose the complexified Lie algebra
  \begin{align*}
  {\mathfrak{sl}}_3({\mathbb{R}}) \otimes_{\mathbb{R}}{\mathbb{C}}\cong (k_0 \otimes{\mathbb{C}}) + (\phi_0\otimes{\mathbb{C}})
  \end{align*}
  as antisymmetric and symmetric parts, then
  \(p_0\otimes{\mathbb{C}}= {\mathbb{C}}X_+ + {\mathbb{C}}X_-\) where
  \(X_{\pm} = { \begin{bmatrix}  {1} & {\pm i} \\  {\pm i} & {-1} \end{bmatrix} }\),
  and \(D_{{\mathrm{CR}}} \phi_f \coloneqq X_- f\).
\end{enumerate}

Conversely, if
\(\phi: {\operatorname{SL}}_2({\mathbb{R}}) \to {\mathbb{C}}\) satisfies
these properties, then \(f(z) = j(g_z, w^\ell)\phi(g_z)\) where
\(g_z \cdot c = z\) is well-defined, holomorphic, weight \(\ell\), level
\(\Gamma\) modular forms.

\end{remark}

\hypertarget{modular-forms-on-mathsfg_2}{%
\subsection{\texorpdfstring{Modular forms on
\({\mathsf{G}}_2\)}{Modular forms on \{\textbackslash mathsf\{G\}\}\_2}}\label{modular-forms-on-mathsfg_2}}

\begin{remark}

Recall that \({\mathsf{G}}_2\) is a simple noncompact Lie group of
dimension 14, with maximal compact
\(K = ({\operatorname{SU}}_2\times {\operatorname{SU}}_2)/\left\langle{\pm I}\right\rangle\).
Write the first factor as \({\operatorname{SU}}_2^l\) for ``long'' and
the second as \({\operatorname{SU}}_2^s\) for ``short'', then the root
system looks like the following:

\includegraphics{figures/2022-03-05_14-36-01.png}

There is an action of \(K\) on
\(V_\ell \coloneqq\operatorname{Sym}^\ell({\mathbb{C}}^2)\otimes\one\),
and the diagonal acts trivially.

\end{remark}

\begin{definition}[Modular forms on $\mathsf G_2$]

Suppose \(\Gamma\leq {\mathsf{G}}_2\) is a congruence subgroup, so
\(\Gamma = {\mathsf{G}}_2({\mathbb{Q}}) \cap K_f\) where
\(K_f \subseteq {\mathsf{G}}_2(M_f)\), and let
\(\ell \in {\mathbb{Z}}_{>0}\),\\
A \textbf{modular form} of weight \(\ell\) and level \(\Gamma\) is a map
\(\phi: {\mathsf{G}}_2 \to V_\ell\) such that

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Growth: \(\phi\) has moderate growth.
\item
  Invariance: \(\phi( \gamma g) = \phi(g)\) for all
  \(\gamma\in \Gamma\).
\item
  Equivariance on a compact: \(\phi(gk) = k^{-1}\phi(g)\) for all
  \(k\in K\).
\item
  Operator equation: \(D_\ell \phi = 0\).
\end{enumerate}

Equivalently, a map
\(\phi: \dcosetl{{\mathsf{G}}_2({\mathbb{Q}})}{{\mathsf{G}}_2({\mathbb{A}})}\)
satisfying similar conditions.

\end{definition}

\begin{remark}

The upshot: modular forms on \({\mathsf{G}}_2\) have a classical Fourier
expansion and Fourier coefficients, which appear very arithmetic.

\end{remark}

\hypertarget{what-is-mathsfg_2}{%
\subsection{\texorpdfstring{What is
\({\mathsf{G}}_2\)?}{What is \{\textbackslash mathsf\{G\}\}\_2?}}\label{what-is-mathsfg_2}}

\begin{remark}

Todos:

\begin{itemize}
\tightlist
\item
  What is \({\mathsf{G}}_2\)?
\item
  What is \(D_\ell\)?
\item
  What are some examples/theorems about modular forms on
  \({\mathsf{G}}_2\)?
\end{itemize}

\end{remark}

\begin{remark}

We'll define a \(C_3{\hbox{-}}\)graded Lie algebra over
\({\mathbb{Q}}\):
\begin{align*}
{\mathfrak{g}}_2 = {\mathfrak{sl}}_3[0] + V_g({\mathbb{Q}})[1] + V_3 {}^{ \vee }({\mathbb{Q}})[2]
,\end{align*}
where \({\mathfrak{sl}}_3\) are the traceless matrices as usual and
\(V_3\) is the 3-dimensional standard representation of
\({\mathfrak{sl}}_3\). The grading will mean that \([x,y]\) will land in
degree
\({\left\lvert {x} \right\rvert} + {\left\lvert {y} \right\rvert}\). The
bracket is defined as follows:
\begin{align*}
[\phi, \phi'] &\coloneqq\phi\phi' - \phi'\phi & \phi,\phi'\in {\mathfrak{sl}}_3 \\
[\phi, v] &\coloneqq\phi(v), & v\in V_3\\
[\phi, \delta] &\coloneqq\phi(\delta),&\delta\in V_3 {}^{ \vee }
.\end{align*}

\end{remark}

\begin{observation}[Constructing $\mathsf G_2$]

\begin{align*}
\bigwedge\nolimits^3 V_3= \one
\implies
\bigwedge\nolimits^2 V_3= V_3 {}^{ \vee }
\implies
\bigwedge\nolimits^2 (V {}^{ \vee }) = V_3
.\end{align*}
Fix a basis \(V_3 = \left\langle{v_1, v_2, v_3}\right\rangle\) and
\(V_3 {}^{ \vee }= \left\langle{\delta_1, \delta_2, \delta_2}\right\rangle\)
its dual basis, then

\begin{itemize}
\tightlist
\item
  \(v_i \vee v_{i+1} = \delta_{i-1}\)
\item
  \(\delta_i \vee\delta_{i+1} = \delta_{i-1}\)
\end{itemize}

Moreover,

\begin{align*}
[v, v'] &= 2 v\vee v' \in \bigwedge\nolimits^2 V_3 \cong V_3 {}^{ \vee }\\
[\delta, \delta'] &= 2\delta\vee\delta' \in \bigwedge\nolimits^2 V_3 {}^{ \vee }\cong V_3 \\
[\delta, v] &= 3v\otimes\delta - \delta(v)\one \in {\mathfrak{sl}}_3
,\end{align*}
noting that the last is traceless and
\(3v\otimes\delta\in V_3\otimes V_3 {}^{ \vee }\cong \mathop{\mathrm{End}}(V_3)\).
All other brackets are determined by antisymmetry and linearity

\end{observation}

\begin{proposition}[Construction of $\mathsf G_2$]

The algebra \({\mathfrak{g}}_2\) as defined above is a simple Lie
algebra, i.e.~the Jacobi identity holds and there are no nontrivial
ideals. Moreover
\begin{align*}
\mathop{\mathrm{Aut}}({\mathfrak{g}}_2) = \left\{{g\in \operatorname{GL}({\mathfrak{g}}_2) {~\mathrel{\Big\vert}~}[gx, gy] = g[x, y]\, \forall x, y \in {\mathfrak{g}}_2}\right\}
\end{align*}
and \({\mathsf{G}}_2 \cong \mathop{\mathrm{Aut}}^0({\mathfrak{g}}_2)\)
is the connected component.

\end{proposition}

\begin{remark}

Note: a similar procedure can be used to define all of the exceptional
groups, see notes.

\end{remark}

\begin{remark}

What is the root diagram for \({\mathfrak{g}}_2\)? Let
\({\mathfrak{h}}\leq {\mathfrak{sl}}_3\) be the diagonal elements, i.e.~
\begin{align*}
{\mathfrak{h}}= \left\{{\sum_{1\leq i\leq 3} \alpha_i E_{ii} {~\mathrel{\Big\vert}~}\sum \alpha_i = 0}\right\}
,\end{align*}
and let \(r_1, r_2, r_3: {\mathfrak{h}}\to {\mathbb{Q}}\) be such that
\begin{align*}
r_j \sum_{1\leq i\leq 3} \alpha_i E_{ii} = \alpha_j
,\end{align*}
i.e.~projection onto the \(j\)th component. Note that \(\sum r_i = 0\).

What are the weights of \({\mathfrak{h}}\) on \({\mathfrak{g}}_2\)?
Since \({\mathfrak{g}}_2 = {\mathfrak{sl}}_3 + V_3 + V_3 {}^{ \vee }\),
the actions are:

\begin{itemize}
\tightlist
\item
  On \(V_3\) it acts by \(r_1, r_2, r_3\).
\item
  On \(V_3 {}^{ \vee }\) it acts by \(-r_1, -r_2, -r_3\).
\item
  On \({\mathfrak{sl}}_3\) it acts by
  \(\left\{{r_i - r_j {~\mathrel{\Big\vert}~}i\neq j}\right\}\).
\end{itemize}

This yields a root diagram:

\includegraphics{figures/2022-03-05_15-01-32.png}

\end{remark}

\begin{remark}

On the differential operator: take the Cartan involution
\begin{align*}
\Theta: {\mathfrak{g}}_2\otimes{\mathbb{R}}\to {\mathfrak{g}}_2 \otimes{\mathbb{R}}
.\end{align*}

Explicitly,

\begin{itemize}
\tightlist
\item
  On \({\mathfrak{sl}}_3\), this acts as \(X\mapsto {}^{-t}X\)
\item
  On \(V_3\), it's \(V_3\mapsto V_3 {}^{ \vee }\) by
  \(v_j\mapsto \delta_j\).
\end{itemize}

Define

\begin{itemize}
\tightlist
\item
  \(k_0 = ({\mathfrak{g}}_2\otimes{\mathbb{R}})^{G=\operatorname{id}}\)
\item
  \(p_0 = ({\mathfrak{g}}_2\otimes{\mathbb{R}})^{G=-\operatorname{id}}\)
\item
  \(K = \left\{{g\in {\mathsf{G}}_2 {~\mathrel{\Big\vert}~}{ \operatorname{Ad} }_g\circ \Theta = \Theta \circ { \operatorname{Ad} }_g}\right\}\)
\item
  \(k = k_0 \otimes{\mathbb{C}}\), something about
  \({\mathfrak{sl}}_3 + {\mathfrak{sl}}_2\)
\item
  \(p = p_0 \otimes{\mathbb{C}}\), something about
  \(V_2\otimes\operatorname{Sym}^3(V_2)\).
\item
  \(D_\ell = {\operatorname{pr}}\tilde D_\ell\), which we'll define.
\end{itemize}

Suppose
\(\phi: {\mathsf{G}}_2\to V_\ell = \operatorname{Sym}^{2\ell}({\mathbb{C}}^2)\otimes\one\)
such that \(\phi(gk) = k^{-1}\phi(g)\) for all \(k\in K\). Let
\(\left\{{X_\alpha}\right\}\) be a basis of \(p\),
\(\left\{{X_\alpha {}^{ \vee }}\right\}\) basis of \(p {}^{ \vee }\),
then
\begin{align*}
\tilde D_\ell \phi = \sum_\alpha X_\alpha p \otimes X_\alpha {}^{ \vee }\in V_\ell \otimes p {}^{ \vee }
.\end{align*}
where \(X_\alpha \phi\) is the derivative of the right regular action,
i.e.~if \(X\in p_0\),
\begin{align*}
(X_p)(g) = {\frac{\partial }{\partial t}\,} \phi(g\exp(tx))\Big|_{t=0}
.\end{align*}
Then
\begin{align*}
V_\ell \otimes\phi {}^{ \vee }
&= (S^{2\ell} \otimes\one) \times V_\ell \boxtimes\operatorname{Sym}^3(V_2) \\
&= (S^{2\ell + 1} + S^{2\ell - 1})\boxtimes S^3(V_2) \\
&\xrightarrow{{\operatorname{pr}}} S^{2\ell -1}(V_\ell) \boxtimes S^3(V_3)
.\end{align*}

This relates

\begin{itemize}
\tightlist
\item
  \({\mathsf{G}}_2\leadsto {\operatorname{SL}}_2\)
\item
  ?
\end{itemize}

\end{remark}

\hypertarget{aaron-pollack-talk-2}{%
\section{Aaron Pollack, Talk 2}\label{aaron-pollack-talk-2}}

\begin{remark}

Last time: modular forms on \({\mathsf{G}}_2\). Note that
\({\mathsf{G}}_2\) over \(K\) does not have a
\({\mathsf{G}}_2{\hbox{-}}\)invariant complex structure, while
\({\operatorname{SL}}_2({\mathbb{R}})/{\operatorname{SO}}_2 = {\mathfrak{h}}\)
has an \({\operatorname{SL}}_2({\mathbb{R}}){\hbox{-}}\)invariant
complex structure.

\end{remark}

\begin{remark}

Today: let \(f(z) = \sum_{k\geq 0} a_f(k)q^k\) of weight \(\ell\) where
\(\phi_f(g) = j(g, i)^{-\ell} f(gi)\) where
\(\phi_f: {\operatorname{SL}}_2({\mathbb{R}}) \to {\mathbb{C}}\). Define
\begin{align*}
W_n: {\operatorname{SL}}_2({\mathbb{R}}) &\to {\mathbb{C}}\\
g &\mapsto j(g, i)^{-\ell} \exp(2\pi i n (gi))
.\end{align*}

Some properties:

\begin{itemize}
\tightlist
\item
  \(W_n\qty{{ \begin{bmatrix}  {1} & {*} \\  {0} & {1} \end{bmatrix} } g} = e^{2\pi i n x}W_n(g)\)
\item
  \(W_n(gk_0) = e^{-i\ell \theta} W_n(g)\) where
  \(k_\theta = { \begin{bmatrix}  {\cos(\theta)} & {-\sin(\theta)} \\  {\sin(\theta)} & {\cos(\theta)} \end{bmatrix} }\),
\item
  \(X_- W_n = 0\)
\item
  \(W_n \operatorname{diag}(y^{1\over 2}, y^{-{1\over 2}}) = y^{\ell\over 2} e^{2\pi i ny}\)
  is complete explicit
\item
  \(\phi_f(g) = \sum a_f(k) W_k(g)\) is the Fourier expansion.
\end{itemize}

\end{remark}

\begin{remark}

What will happen: we'll define
\(\phi: \dcosetl{\Gamma}{{\mathsf{G}}_2}\to V_\ell\) where
\(V_\ell = \operatorname{Sym}^2({\mathbb{C}}^2)\otimes\one\) which
admits an action by
\(K = {\operatorname{SU}}_2\times {\operatorname{SU}}_2/\pm I\). In this
case, we'll essentially have
\(\phi \approx \sum_{f\in ?} a_\phi(f) W_f(g)\) where the
\(a_\pi(f) \in {\mathbb{C}}\) are Fourier coefficients and \(W_f\)
satisfies similar properties.

\end{remark}

\begin{remark}

Recall that
\({\mathfrak{g}}_2 = {\mathfrak{sl}}_3 + V_3 + V_3 {}^{ \vee }\),
spanned by
\(\left\{{E_{ij}}\right\}, \left\{{v_1,v_2,v_3}\right\}, \left\{{\delta_1, \delta_2, \delta_3}\right\}\)
respectively.

Note that

\begin{itemize}
\tightlist
\item
  \({\mathsf{G}}_2\) has 2 conjugacy classes of maximal parabolics,
\item
  \(P\) will be the parabolic where \(\mathsf{Lie}(P)\) yields the top 3
  layers of the root diagram
\item
  \(P = MN\) where \(M\cong \operatorname{GL}_2\) and
  \(N\supseteq Z = [N, N]\) with \(N/Z\) abelian.
\end{itemize}

We want to define a Fourier expansion along the unipotent radical of
\(P\).

\end{remark}

\begin{remark}

Some facts:

\begin{itemize}
\tightlist
\item
  \(Z = \exp({\mathbb{R}}E_{13})\)
\item
  \(W = {\mathbb{R}}E_{12} + {\mathbb{R}}v_1 + {\mathbb{R}}\delta_3 + {\mathbb{R}}E_{23}\)
\item
  \(N/Z = \exp(W)\)
\item
  \(M\curvearrowright Z\) by the determinant
\item
  \(M\curvearrowright N/Z\) as
  \(\operatorname{Sym}^3(V_3) \otimes\operatorname{det}(V_3)^{-1}\)
\item
  There is a symplectic form on \(W\) where
  \([w, w'] = {\left\langle {w},~{w'} \right\rangle} E_{13}\)
\item
  Explicitly, one can write
  \(w = \sum a E_{12} + {b\over 3} v_1 + {c\over 3} \delta_3 + d E_{13}\)
  and \(w'\) similarly, then
  \({\left\langle {w},~{w'} \right\rangle} = ad' - {bc'\over 3} + {cb'\over 3}- da'\),
  and
  \({\left\langle {mw},~{mw'} \right\rangle} = \operatorname{det}(m) {\left\langle {w},~{w'} \right\rangle}\).
\end{itemize}

\end{remark}

\begin{remark}

What are the characters of \(N\)? Suppose

\begin{itemize}
\tightlist
\item
  \(\phi\) is an automorphic form on \({\mathsf{G}}_2({\mathbb{A}})\)
\item
  \(\psi: \dcosetl{{\mathbb{Q}}}{{\mathbb{A}}}\to {\mathbb{C}}^{\times}\)
  is a fixed adelic character
\item
  \(w\in W({\mathbb{Q}})\)
\end{itemize}

Define
\begin{align*}
\phi_w(g) \coloneqq\int_{[N]} \psi^{-1}({\left\langle {w},~{\mkern 1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern 1.5mu} \right\rangle}) \phi(ng) \,dn
,\end{align*}
where \(\mkern 1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern 1.5mu\)
is the image of \(n\) in \(N/Z\) which we identify with \(W\) via the
exponential. Similarly define
\begin{align*}
\phi_Z(g) = \int_{[Z]} \phi(zg)\,dz,\qquad
\phi_N(g) = \int_{[N]} \phi(zg)\,dz
.\end{align*}
Then
\begin{align*}
\phi_Z(g) = \phi_N(g) + \sum_{w\in W({\mathbb{Q}})} \phi_w(g)
,\end{align*}
and we'll produce a refinement.

\end{remark}

\begin{proposition}[?]

\begin{align*}
\phi_Z(g) \equiv 0 \implies \phi(g) \equiv 0
.\end{align*}

\end{proposition}

\hypertarget{generalized-whittaker-functions}{%
\subsection{Generalized Whittaker
Functions}\label{generalized-whittaker-functions}}

\begin{definition}[Generalized Whittaker functions]

Suppose
\(\phi: \dcosetl{{\mathsf{G}}_2({\mathbb{Q}})}{{\mathsf{G}}_2({\mathbb{A}})} \to V_\ell\)
is a modular form of weight \(\ell\). These satisfy

\begin{itemize}
\tightlist
\item
  \(\phi_w(g)\) is of moderate growth.
\item
  \(\phi_w(g) = \psi({\left\langle {w},~{\mkern 1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern 1.5mu} \right\rangle}) \psi_W(g)\)
  (equivariant for the Heisenberg parabolic)
\item
  \(\phi_w(gk) = k^{-1}\phi_w(g)\) (equivariant for \(K\))
\item
  \(D_\ell \phi_w = 0\).
\end{itemize}

Call such functions satisfying these properties \textbf{general
Whittaker functions of type \((w ,\ell)\)}.

\end{definition}

\begin{remark}

We'll show that such functions are uniquely determined up to a scalar
multiple, i.e.~for some explicit \(W_w\),
\begin{align*}
\phi_w(g) = \lambda W_w(g)
.\end{align*}
From this, we'll obtain a Fourier expansion for \(\phi\) a modular form
of weight \(\ell\):
\begin{align*}
\phi_Z(g) = \phi_N(g) + \sum_{w\neq 0} a_{\phi}(w) W_w(g)
.\end{align*}

\end{remark}

\begin{remark}

Identify \(W\) as a space \(B\) of binary cubics under
\begin{align*}
W &\to B \\
w \coloneqq a E_{12} + {b\over 3} v_1 + {c\over 3} \delta_3 + d E_{23}
&\mapsto
f_w\coloneqq
au^3 + bu^2v + cuv^2 + dv^3
.\end{align*}
For \(w\in W({\mathbb{R}})\setminus\left\{{0}\right\}\), for
\(m\in \operatorname{GL}_2({\mathbb{R}})\) define
\begin{align*}
\beta_w(m) \coloneqq{\left\langle {w},~{m\cdot (u-? v)^3} \right\rangle}
,\end{align*}
which will appear in Fourier expansions.

\end{remark}

\begin{proposition}[?]

TFAE:

\begin{itemize}
\tightlist
\item
  \(\beta_w(m) \neq 0\) for all
  \(m\in \operatorname{GL}_2({\mathbb{R}})\),
\item
  \(f_w(z, 1) \neq 0\) for \(z\in {\mathfrak{h}}\),
\item
  \(f_w\) splits into linear factors over \({\mathbb{R}}\).
\end{itemize}

\end{proposition}

\begin{definition}[PSD]

If \(w\) satisfies these properties, say \(w\) is \textbf{positive
semidefinite} and write \(w\geq 0\).

\end{definition}

\begin{example}[of PSD binary cubics]

\envlist

\begin{itemize}
\tightlist
\item
  \(f_w(u, v) \coloneqq au^3 \geq 0\).
\item
  \(f_w(u, v) \coloneqq-u^3 + uv^2 = u(v-u)(v+u)\geq 0\).
\item
  \(f_w(u, v) \coloneqq u^3 + v^2 \not\geq 0\).
\end{itemize}

\end{example}

\begin{definition}[?]

For \(m\in \operatorname{GL}_2({\mathbb{R}}) = M({\mathbb{R}})\) and
\(w\geq 0\) PSD,
\begin{align*}
W_w(m) = {\left\lvert {\operatorname{det}w} \right\rvert}\operatorname{det}(w)^\ell \sum_{-\ell \leq v\leq 0}
\qty{{\left\lvert {\beta_w(m)} \right\rvert} \over \beta_w(m) }^v K_v({\left\lvert {\beta_w(m)} \right\rvert}) {x^{\ell+v} y^{\ell -v} \over (\ell+v)! (\ell-v)!}
.\end{align*}
where \(x,y\) are a fixed basis of \(V_\ell\), and \(K_v\) is a
classical \(K{\hbox{-}}\)Bessel function
\begin{align*}
K_v(y) \coloneqq{1\over ?}\int_0^N e^{-{y(t+t^{-1})\over 2 }} t^v {\,dt\over t}
,\end{align*}
which diverges at \(y=0\).

\end{definition}

\begin{remark}

These functions \(W_w: M({\mathbb{R}}) \to V_\ell\) extend uniquely to
\({\mathsf{G}}_2\to V_\ell\), viz

\begin{itemize}
\tightlist
\item
  \(W_w(ng) = e^{2\pi i {\left\langle {w},~{\mkern 1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern 1.5mu} \right\rangle}} W_w(g)\)
  for all \(n\in N({\mathbb{R}})\)
\item
  \(W_w(gk) = k^{-1}W_w(g)\) for all \(k\in K\).
\end{itemize}

\end{remark}

\begin{theorem}[?]

Suppose \(w\neq 0\) and \(F\) is a generalized Whittaker function of
type \((w, \ell)\). Then

\begin{itemize}
\tightlist
\item
  \(w\not\geq 0\implies F = 0\)
\item
  \(w\geq 0\implies F(g) = \lambda W_w(g)\) for some
  \(\lambda \in {\mathbb{C}}\)
\end{itemize}

Consequently, if \(\phi\) is a modular form on \({\mathsf{G}}_2\) of
weight \(\ell\), there exist \(a_\phi(w)\in {\mathbb{C}}\) with
\begin{align*}
\phi_Z(g) = \phi_N(g) + \sum_{w\geq 0 \text{ integral}} a_\phi(w) W_w(g)
.\end{align*}
Moreover, \(\phi_N\) can be explicitly described in terms of modular
forms of weight \(3\ell\) on \(\operatorname{GL}_2\).

\end{theorem}

\begin{definition}[?]

The terms \(a_\phi(w)\) are by definition the Fourier coefficients of
\(\phi\).

\end{definition}

\begin{remark}

Gan-Gross-Savim used a multiplicity 1 result of Wallach to define the
Fourier coefficients without using the explicit function \(W_w(s)\).

\end{remark}

\hypertarget{aaron-pollack-talk-3-examples-of-and-theorems-about-modular-forms-on-mathsfg_2}{%
\section{\texorpdfstring{Aaron Pollack, Talk 3: Examples of (and
theorems about) modular forms on
\({\mathsf{G}}_2\)}{Aaron Pollack, Talk 3: Examples of (and theorems about) modular forms on \{\textbackslash mathsf\{G\}\}\_2}}\label{aaron-pollack-talk-3-examples-of-and-theorems-about-modular-forms-on-mathsfg_2}}

\hypertarget{degenerate-eisenstein-series}{%
\subsection{Degenerate Eisenstein
Series}\label{degenerate-eisenstein-series}}

\begin{remark}

Recall \({\mathsf{G}}_2\supseteq P\) a Heisenberg parabolic, with
\(P = MN\) where \(M\cong \operatorname{GL}_2\). Write \(\nu\) for the
composition
\(P\to M \xrightarrow{\operatorname{det}} \operatorname{GL}_1\). Suppose
\(\ell > 0\) is even, and recall that
\(V_\ell = \operatorname{Sym}^2({\mathbb{C}}^2)\otimes\curvearrowleft K \leq {\mathsf{G}}_2\)
for \(K\) a maximal compact. Let
\begin{align*}
f_{\ell, \infty} (g; s) = \operatorname{Ind}_{P({\mathbb{R}})}^{{\mathsf{G}}_2({\mathbb{R}})} {\left\lvert {\nu} \right\rvert}^s \otimes V_\ell
\end{align*}
be defined by

\begin{align*}
f_{\ell, \infty}(pg^j s) &= N(\mu)^s f_{e\ll, \infty} (g; s) \quad \forall p\in P({\mathbb{R}}) \\
f_{\ell,\infty}(gk; s) &= k^{-1}f_{\ell, \infty}(g) \quad \forall k\in K
.\end{align*}

By the Iwasawa decomposition \(C_3({\mathbb{R}}) = P({\mathbb{R}}) K\),
\(f\) is uniquely determined one we set
\begin{align*}
f_{\ell, \infty}(1) = x^\ell y^\ell \in V_\ell = \left\langle{x^?, x^?, \cdots, y^{2\ell}}\right\rangle
.\end{align*}
Let \(f_?\) be a flat section in
\(\operatorname{Ind}_{P({\mathbb{A}}_f)}^{{\mathsf{G}}_2({\mathbb{A}}_f)}({\left\lvert {\nu} \right\rvert}^3)\),
and let
\(f_g(g, s) = f_?(gf, s) f_{\ell,\infty}(g_?; s) \in {\mathsf{G}}_2({\mathbb{A}})\).
Define
\begin{align*}
E_\ell(g,f,s) = \sum_{\gamma\in \dcosetl{P({\mathbb{Q}})}{ {\mathsf{G}}_2({\mathbb{Q}})} } f_\ell(\gamma g, s)
.\end{align*}
If \(\Re(s) > 3\), set \(E_\ell(g) \coloneqq E_\ell(g, f, s=\ell+1)\).

\end{remark}

\begin{theorem}[?]

If \(\ell>0\) is even and \(\ell\geq 4\), then \(E_\ell(g)\) is a
quaternionic modular form on \({\mathsf{G}}_2\) of weight \(\ell\).

\end{theorem}

\begin{proof}[?]

\(f_{\ell, ?}(g, s=\ell+1)\) is annihilated by \(D_\ell\), so
\(E_\ell(g)\) is as well by absolute convergence.

\end{proof}

\begin{remark}

If \(\pi\) is a cuspidal automorphic representation of
\(\operatorname{GL}_2 = M\) associated to a holomorphic weight \(3\ell\)
modular form which is cuspidal,

\begin{itemize}
\tightlist
\item
  \(f_\pi\in \operatorname{Ind}_{P({\mathbb{A}})}^{{\mathsf{G}}_2({\mathbb{A}})}(\pi)\)
\item
  \(E(g, f_\pi) = \sum_{ \gamma\in \dcosetl{P({\mathbb{Q}})}{{\mathsf{G}}_2({\mathbb{Q}})} } f_\pi( \gamma g)\)
\item
  If \(\ell \geq 6\) this is a weight \(\ell\) modular form on
  \({\mathsf{G}}_2\).
\item
  If \(\ell=4\), one can make sense of \(E(g, f_\pi)\) using analytic
  continuation to produce a weight 4 modular form on \({\mathsf{G}}_2\)
  associated to the Ramanujan \(\Delta\).
\end{itemize}

\end{remark}

\begin{fact}

If \(\phi\) is a level 1 quaternionic modular form on
\({\mathsf{G}}_2\),

\begin{itemize}
\tightlist
\item
  \(a_\phi(w) \neq 0 \implies f_w(u, v)= au^3 + \cdots + dv^3\) with
  \(a,\cdots,d\in {\mathbb{Z}}\) (integrality)
\item
  \(a_p(w\gamma) = \operatorname{det}(\gamma)^\ell a_\phi(w)\) for
  \(r\in \operatorname{GL}_2({\mathbb{Z}})\), so Fourier coefficients
  are constant on orbits of binary cubic forms.
\end{itemize}

\end{fact}

\begin{fact}

There is a bijection
\begin{align*}
\left\{{\text{Integral binary cubic forms}}\right\}/\operatorname{GL}_2({\mathbb{Z}})
\rightleftharpoons
\left\{{\text{Cubic rings}}\right\}{_{\scriptstyle / \sim} }
,\end{align*}
where cubic rings are free rank 3 \({\mathbb{Z}}{\hbox{-}}\)algebras.
Thus if \(\ell>0\) is even, \(\phi\) is a level 1 weight \(\ell\)
modular form on \({\mathsf{G}}_2\), and \(A\) is a cubic ring, there is
a well-defined map \(a_\phi(A) = a_\phi(w)\) if
\(A\rightleftharpoons f_w\).

\end{fact}

\begin{remark}

If \(f_w\) is nondegenerate, the cubic ring \(A(f_w)\) associated to
\(f_w\) is totally real \(\iff f\) is positive semidefinite.

\end{remark}

\begin{theorem}[?]

Suppose \(A\) is the maximal order in a totally real cubic etale
\({\mathbb{Q}}{\hbox{-}}\)algebra \(E\). There exists a constant
\(c_\ell\in {\mathbb{C}}\), independent of \(A\), such that
\begin{align*}
a_{E_\ell}(A) = c_\ell \zeta_E(1-\ell)
,\end{align*}
where \(\ell\) is even. The LHS are Fourier coefficients of modular
forms on \({\mathsf{G}}_2\).

\end{theorem}

\begin{remark}

It is not known that \(c_\ell\) is nonzero.

\end{remark}

\begin{question}

An open question: \(E(g, f_\pi)\) is Eisenstein, can anything be said
about its Fourier coefficients.

\end{question}

\hypertarget{cusp-forms-1}{%
\subsection{Cusp Forms}\label{cusp-forms-1}}

\begin{theorem}[?]

Suppose \(\ell\geq 16\) is even. There exist nonzero cusp forms on
\({\mathsf{G}}_2\) of weight \(\ell\), all of whose Fourier coefficients
are algebraic integers.

\end{theorem}

\begin{proof}[of theorem]

Steps:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Start with a holomorphic Siegel modular form \(f\) on
  \({\operatorname{SP}}_4\) of weight \(\ell\), so \(f\) has Fourier
  coefficients in
  \({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Z} \mkern-1.5mu}\mkern 1.5mu }\)
\item
  Take a \(G{\hbox{-}}\)lift of \(f\) to \({\operatorname{SO}}_{4,4}\)
  to obtain \(G(f)\), and define
  \begin{align*}
  \Theta(f)(g) = \int_{[{\operatorname{SP}}_4]} \theta(g, h) \mkern 1.5mu\overline{\mkern-1.5muf(h)\mkern-1.5mu}\mkern 1.5mu \, dh
  ,\end{align*}
  then \(\Theta\) on
  \({\operatorname{SO}}_{4,4}\times {\operatorname{SP}}_4\) is a
  \(\theta\) function.
\item
  There is a good theory of quaternionic modular forms on
  \({\operatorname{SO}}_{4, n}\), so choose \(\theta(g,h)\) such that
  \(\Theta(f)\) is a one of weight \(\ell\) (and cuspidal).
\item
  Express the Fourier coefficients of \(\Theta(f)\) in terms of
  classical Fourier coefficients of \(f\), showing that the Fourier
  coefficients of \(G(f)\) are in
  \({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Z} \mkern-1.5mu}\mkern 1.5mu }\).
\item
  Use
  \({\mathsf{G}}_2 \xhookrightarrow{\iota}{\operatorname{SO}}_{4,4}\)
  and pullback to obtain \(i^*(\Theta(f))\), which is still cuspidal and
  has Fourier coefficients that are sums of the original coefficients,
  so still in
  \({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Z} \mkern-1.5mu}\mkern 1.5mu }\).
\end{enumerate}

\end{proof}

\begin{theorem}[R. Dalal]

There is an explicit dimension formula for the level 1 cuspidal
quaternionic modular forms of weight \(\ell\). In particular, the
smallest is a level 1 cusp form of weight 6.

\end{theorem}

\begin{theorem}[Cicek-Dadivdoff,Dijok,Hammonds, P, Roy]

Suppose \(\phi\) is a level 1 cuspidal quaternionic modular form on
\({\mathsf{G}}_2\) associated to a cuspidal automorphic representation
\(\pi\) on \({\mathsf{G}}_2({\mathbb{A}})\). Suppose that the Fourier
coefficient
\(a_\phi({\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{3} } }) \neq 0\),
then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  The complete standard \(L{\hbox{-}}\)function of \(\pi\) has a
  functional equation:
  \begin{align*}
  \Lambda(\pi, \text{std}, s) = \Lambda(\pi, \text{std}, 1-s)
  .\end{align*}
\item
  There exists a Dirichlet series for this \(L{\hbox{-}}\)function
  expressing the Fourier coefficients in terms of an
  \(L{\hbox{-}}\)function:
  \begin{align*}
  \sum_{T \subseteq {\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{3} }  },n\geq 1} {a_\phi({\mathbb{Z}}+ nT) \over [{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{3} }  } : T]^{s-\ell+1}  }n^{-s}
  =
  a_p({\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{3} }  }) {L(\pi, \text{std}, s-z\ell+1) \over \zeta(s-2\ell + 2)^2 \zeta(2s-4\ell+2) }
  .\end{align*}
\end{enumerate}

\end{theorem}

\begin{proof}[?]

Carry out a refined analysis of a Rankin-Selberg integral (due to
Gurevich-Segal).

\end{proof}

\hypertarget{a-theorem}{%
\subsection{A Theorem}\label{a-theorem}}

\begin{remark}

There is a theory of \emph{half}-integral weight modular forms on
\({\mathsf{G}}_2\). These have a good notion of Fourier coefficients
taking values in \({\mathbb{C}}/\left\{{\pm 1}\right\}\).

Suppose \(R \subseteq E\) is a cubic ring in a totally real cubic field.
Let \({\partial}_R\) be the different, and let \(Q_R\) be the square
roots of \({\partial}_R^{-1}\) in the narrow class group of \(E\). Say
\((I, \mu)\) is balanced

\begin{itemize}
\tightlist
\item
  \(I\) is a fractional ideal in \(R\).
\item
  \(\mu\in E_{>0}^{\times}\) is totally positive
\item
  \(I \mu^2 \subseteq \partial_R^{-1}\)
\item
  \(N(I)^2 N(\mu) {\operatorname{disc}}(R) = 1\).
\end{itemize}

Note that if \(R\) is the maximal order, \((I, \mu)\) is balanced iff
\(I^2\mu = {\partial}_R^{-1}\). Define an equivalence relation by
\begin{align*}
(I, \mu) \sim (I', \mu') \iff \exists \beta\in E^{\times}, I' = \beta I, \mu' = \beta^{-2}\mu
\end{align*}
and set \(Q_R\) to be the balanced pairs mod equivalence.

\end{remark}

\begin{remark}

\envlist

\begin{itemize}
\tightlist
\item
  \(Q_R\) can be empty
\item
  For \(Q_R\) nonempty and \(R\) a maximal order in \(E\),
  \begin{align*}
  {\left\lvert {Q_R} \right\rvert} = {\sharp}{ \operatorname{Cl}} E^{\times}[2]
  .\end{align*}
\end{itemize}

\end{remark}

\begin{theorem}[Leslie-P]

There exists a weight \(1/2\) modular form \(\theta'\) on
\({\mathsf{G}}_2\) whose Fourier coefficients include the numbers
\(\pm {\left\lvert {Q_R} \right\rvert}\) for \(R\) \emph{even
monogenic}, i.e.~
\begin{align*}
R = {\mathbb{Z}}[y]/ \left\langle{y^3 + cy^2 + by+a, a,b,c\in 2{\mathbb{Z}}}\right\rangle
.\end{align*}

\end{theorem}

\begin{proof}[?]

Define \(\Theta\) on \(\tilde {\operatorname{F}}_4\)? Then let
\(\Theta'\) be a pullback along
\({\mathsf{G}}_2\to {\operatorname{F}}_3\).

\end{proof}

\hypertarget{aaron-pollack-talk-4-beyond-mathsfg_2}{%
\section{\texorpdfstring{Aaron Pollack, Talk 4: Beyond
\({\mathsf{G}}_2\)}{Aaron Pollack, Talk 4: Beyond \{\textbackslash mathsf\{G\}\}\_2}}\label{aaron-pollack-talk-4-beyond-mathsfg_2}}

\begin{remark}

Upshot for today: there exist groups
\({\mathsf{G}}_3, {\operatorname{F}}_4, {\mathbf{E}}_{n, 4}\) for
\(n=6,7,8\) where \({\mathsf{G}}_2\) is split and the \({\mathbf{E}}\)
groups are rank 4 over \({\mathbb{R}}\). These admit modular forms with
Fourier expansions and coefficients similar to the \({\mathsf{G}}_2\)
story. We'll define these exceptional groups today.

\end{remark}

\hypertarget{exceptional-algebras}{%
\subsection{Exceptional Algebras}\label{exceptional-algebras}}

\begin{remark}

Let be a composition algebra over \(k\), where
\(\operatorname{ch}k = 0\), with a multiplication
\(C{ {}^{ \scriptstyle\otimes_{k}^{2} } }\to C\) which is not
necessarily commutative or associative. There exists a norm map
\(n_C: C\to K\) given by a nondegenerate quadratic form with
\(n_C(xy) = n_C(x) n_C(y)\).

\end{remark}

\begin{example}[?]

\envlist

\begin{itemize}
\item
  \(C = k\) and
  \begin{align*}
  n_c: k &\to k \\
  x&\mapsto x^2
  .\end{align*}
\item
  \(C = E/k\) for \(E\) a quadratic etale extension,
  \(n_C = \operatorname{Nm}_{E/k}\).
\item
  \(C = B/k\) for \(B\) a quaternion algebra with
  \(n_C = n_{B, { \text{red} }}\)
\item
  \(C = \Theta\) an octonion algebra, with \(\Theta = B \bigoplus B\).
  There is an involution \(C\to C\) with

  \begin{itemize}
  \tightlist
  \item
    \(x + x^* = \operatorname{Trace}_C(x) 1\in k1\),
  \item
    \(xx^* = n_C(x)1\in k1\)
  \end{itemize}
\end{itemize}

\end{example}

\begin{definition}[?]

Let \(J_C = H_3(C)\) be Hermitian \(3\times 3\) matrices with
coefficients in \(C\), so
\begin{align*}
J_C = \left\{{ 
\begin{bmatrix}
 c_1 & x_3 & x_2^* \\
 x_3^* & c_2 & x_1 \\
 x_2 & x_1^* & c_3
\end{bmatrix}
 {~\mathrel{\Big\vert}~}c_i\in k, c_i\in C
}\right\}
.\end{align*}
This has dimension \(3+3C\) over \(k\).

\end{definition}

\begin{example}[?]

For \(C=k, H_3(K)\) are symmetric \(3\times 3\) matrices and there is a
determinant map
\begin{align*}
\operatorname{det}: J_C &\to k \\
X&\mapsto c_1c_2 c_3 - \sum c_i n_C(x_i) + \operatorname{Trace}_C(x_1(x_2 x_3))
.\end{align*}

If \(C=k\) this is the usual determinant. Note that
\(M_J' = \left\{{g\in \operatorname{GL}(J_C) {~\mathrel{\Big\vert}~}\operatorname{det}(gX) = \operatorname{det}(X) \forall X\in J_C}\right\}\)
has positive dimension, and is thus infinite, making it an interesting
algebra.

\end{example}

\begin{remark}

Idea: there exists a group \(G_{J_C}\) such that

\begin{itemize}
\tightlist
\item
  \(C = {\mathbb{Q}}\leadsto {\operatorname{F}}_4\),
\item
  \(C = K\) quadratic imaginary \(\leadsto {\mathbf{E}}_{6,4}\),
\item
  \(C= B\) a quaternionic algebra \(\leadsto {\mathbf{E}}_{7,4}\),
\item
  \(C = \Theta\) an octonionic algebra \(\leadsto {\mathbf{E}}_{8, 4}\).
\end{itemize}

All have a good notion of quaternionic modular forms and Fourier
expansions/coefficients.

A degree map \(J_C \xrightarrow{\deg} k\) commuting with
\(x\mapsto x^3\) and \(\operatorname{det}\) recovers
\(G_{J_C = k} = {\mathsf{G}}_2\). Recall
\begin{align*}
{\mathfrak{g}}_2 = {\mathfrak{sl}}_{2, \ell}[0] + {\mathfrak{sl}}_{2, s}[0] + V_2 \otimes W[1]
\end{align*}
which is \(C_2{\hbox{-}}\)graded; we'll mimic this to construct
\({\mathfrak{g}}_{J_C}\).

\end{remark}

\hypertarget{freudenthal-construction}{%
\subsection{Freudenthal Construction}\label{freudenthal-construction}}

\begin{definition}[Quaternionic exceptional groups]

Let \(J = J_C/{\mathbb{Q}}\) and \(k={\mathbb{Q}}\) and
\begin{align*}
W_J = {\mathbb{Q}}\bigoplus J \oplus J {}^{ \vee }\oplus {\mathbb{Q}}
.\end{align*}
There is a symplectic form
\begin{align*}
{\left\langle {{\left[ {a,b,c,d} \right]}},~{{\left[ {a,b,c,d} \right]}} \right\rangle} = ad' - (b,c') - (c, b') - dc'
.\end{align*}
There is a degree 4 polynomial map \(q: W_J\to {\mathbb{Q}}\). Define
\begin{align*}
H_J^1 = \left\{{g\in \operatorname{GL}(W_J) {~\mathrel{\Big\vert}~}{\left\langle {gw},~{gw'} \right\rangle} = {\left\langle {w},~{w'} \right\rangle} \,\forall w,w'\in W_J,\, q(gw) = q(w)}\right\}
.\end{align*}
This recovers:

\begin{longtable}[]{@{}lll@{}}
\toprule
\(C\) & \(H_J^1\) & \({\mathfrak{g}}_J\) \\
\midrule
\endhead
\({\mathbb{Q}}\) & \(\operatorname{C}_3\) & \({\operatorname{F}}_4\) \\
\(K\) & \(\operatorname{A}_5\) & \({\mathbf{E}}_6\) \\
\(B\) & \({ \mathsf{D} }_6\) & \({\mathbf{E}}_7\) \\
\(\Theta\) & \(E_7\) & \({\mathbf{E}}_8\) \\
\bottomrule
\end{longtable}

Define
\begin{align*}
{\mathfrak{g}}_J ={\mathfrak{sl}}_2[0] + {\mathfrak{h}}_J^0[0] + (V_2\otimes W_J)[1]
,\end{align*}
where \({\mathfrak{h}}_J^0 = \mathsf{Lie}(H_J^1)\), and define
\begin{align*}
G_J \coloneqq\mathop{\mathrm{Aut}}^0({\mathfrak{g}}_J)
.\end{align*}

If \(n_C: C\otimes{\mathbb{R}}\to {\mathbb{R}}\) is positive definite,
we say \({\mathsf{G}}_J\) is a \textbf{quaternionic exceptional group}.

\end{definition}

\begin{fact}

If \(K_J \subseteq {\mathsf{G}}_J({\mathbb{R}})\) is a maximal compact,
then
\begin{align*}
K_J = {{\operatorname{SU}}_2 \times L_J' \over \mu_2}
\end{align*}
where \(L_J^1\) is a compact form of \(H_J^1\).

There is a Cartan involution
\(\theta: {\mathfrak{g}}_J\to {\mathfrak{g}}_J\), which over
\({\mathbb{C}}\) yields
\begin{align*}
{\mathfrak{g}}_J^{\theta = \operatorname{id}} &= k_0\otimes{\mathbb{C}}\cong {\mathfrak{sl}}_2 + {\mathfrak{h}}_J^0  \\
{\mathfrak{g}}_J^{\theta = -\operatorname{id}} &= p_0\otimes{\mathbb{C}}\cong V_2 + W_J  
.\end{align*}

\end{fact}

\begin{remark}

There is an action
\(K_J\curvearrowright V_\ell = \operatorname{Sym}^{2\ell}({\mathbb{C}}^2)\otimes\one\)

\end{remark}

\begin{definition}[Modular forms]

A \textbf{modular form} on \({\mathsf{G}}_3\) of weight \(\ell\) is an
automorphic form
\begin{align*}
\phi: \dcosetl{G_J({\mathbb{R}})}{G_J({\mathbb{A}})} \to V_\ell
\end{align*}
such that

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\phi(gk) = k^{-1}\phi(g)\) for all \(k\in K_J\)
\item
  \(D_\ell \phi \equiv 0\)
\end{enumerate}

Here \(D_\ell\) is defined as in the \({\mathsf{G}}_2\) case, replacing
\(\operatorname{Sym}^3(V_2) = W\) with \(W_J\).

\end{definition}

\begin{remark}

There is a Heisenberg parabolic \(P = MN \leq G_J\) with \(M = H_J\),
and \(N \supseteq Z\) a two-step filtration with \(Z\) 1-dimensional and
\(N/Z \cong W_J\) abelian.

\end{remark}

\begin{theorem}[?]

Modular forms on \(G_J\) of weight \(\ell\)

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Have Fourier coefficients and expansions along \(N/Z\):
  \begin{align*}
  \phi_Z(g) = \phi_N(g) + \sum_{w\in W_J({\mathbb{Q}}), w\geq 0} a_\phi(w) W_w(g)
  ,\end{align*}
  where \(a_\phi(w)\in {\mathbb{C}}\) are the Fourier coefficients of
  \(\phi\) and \(W_w\) are completely explicit.
\item
  Under appropriate embeddings
  \begin{align*}
  {\mathsf{G}}_2 \hookrightarrow{\operatorname{F}}_4 \hookrightarrow{\mathbf{E}}_{6,4}\hookrightarrow E_{7,4} \hookrightarrow E_{8,4}
  ,\end{align*}
  for a modular form \(\phi\) of weight \(\ell\) of one groups, the
  pullbacks \(i^* \phi\) to a smaller group are again modular forms of
  weight \(\ell\) whose Fourier coefficients are sums of Fourier
  coefficients of \(\phi\).
\end{enumerate}

\end{theorem}

\begin{theorem}[?]

\envlist

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  There exists a nonzero weight 4 modular form \(\theta_{\min}\) on
  \({\mathbf{E}}_{8, 4}\) with rational Fourier coefficients.
\item
  There exists a nonzero weight 8 modular form \(\tilde\theta_{\min}\)
  on \({\mathbf{E}}_{8, 4}\) with rational Fourier coefficients.
\end{enumerate}

\end{theorem}

\begin{proof}[Sketch]

\envlist

\begin{itemize}
\tightlist
\item
  Construct \(\tilde\theta_{\min}\) using Eisenstein series
\item
  Savim: most of the Fourier coefficients are zero, particularly the
  ones that are harder to compute explicitly
\item
  By explicit computations, the remaining coefficients are rational.
\end{itemize}

\end{proof}

\begin{definition}[?]

Say a modular form \(\phi\) on \({\mathsf{G}}_3\) is
\textbf{distinguished} iff

\begin{itemize}
\tightlist
\item
  There exists a \(w_0\in W_J({\mathbb{Q}})\) such that \(q(W_0)\neq 0\)
  and \(a_\phi(w_0) \neq 0\)
\item
  If \(w\in W_J({\mathbb{Q}}), a_\phi(w)\neq 0\), then
  \(q(w)\equiv q(w_0) \operatorname{mod}({\mathbb{Q}}^{\times}){ {}^{ \scriptscriptstyle\times^{2} } }\).
\end{itemize}

\end{definition}

\begin{theorem}[?]

Suppose \(K/{\mathbb{Q}}\) is quadratic imaginary, then there exists a
distinguished modular form of weight 4 \(\Theta_K\) on
\(G_{J_K} = {\mathbf{E}}_{6,4}\).

\end{theorem}

\begin{proof}[?]

Set \(\Theta_K i^*(\Theta_{\min})\), pullback to \({\mathbf{E}}_{6,4}\).
By arithmetic invariant theory, one shows it is distinguished.

\end{proof}

\hypertarget{zhiwei-yun-rigidity-method-for-automorphic-forms-over-function-fields-lecture-1}{%
\section{Zhiwei Yun: Rigidity method for automorphic forms over function
fields (Lecture
1)}\label{zhiwei-yun-rigidity-method-for-automorphic-forms-over-function-fields-lecture-1}}

\begin{remark}

Goal: construct automorphic data over function fields and working out
the Langlands correspondence for such examples. Setup:

\begin{itemize}
\item
  \(k= {\mathbb{F}}_q\),
\item
  \(X_{/ {k}}\) a projective, smooth, geometrically connected algebraic
  curve.\footnote{This is already interesting in the case of
    \(X = {\mathbb{P}}^1\).}
\item
  \(F\coloneqq k(X)\) its function field, noting that
  \(\operatorname{trdeg}F = 1\).
\item
  \({\left\lvert {X} \right\rvert} \coloneqq{\operatorname{Places}}(X)\),
  and \({\mathcal{O}}_x \twoheadrightarrow k_x\) with
  \(F_x \subseteq {\mathcal{O}}_x\),
  \(F_x = k_x{\left(\left( t_x \right)\right) }\) will be formal Laurent
  series and
  \({\mathcal{O}}_x \cong k{\left[\left[ t_x \right]\right] }\).
\item
  \(G_{/ {k}}\) a split semisimple group,
  e.g.~\(G={\operatorname{SL}}_n, \operatorname{PGL}_n, {\mathsf{Sp}}_{2n}, {\mathsf{G}}_2, {\mathbf{E}}_8,\cdots\)
\item
  The adeles
  \({\mathbb{A}}= \prod^{\operatorname{res}}_{x\in {\left\lvert {X} \right\rvert}} F_x\),
  where the \(F_x\) will be nonarchimedean fields,
\item
  \(G({\mathbb{A}}) = \prod^{\operatorname{res}}_{x\in {\left\lvert {X} \right\rvert}} G(F_x)\)
  where almost all components are in \(G({\mathcal{O}}_x)\).
\item
  Level groups:

  \begin{itemize}
  \tightlist
  \item
    \(K = \prod^{\operatorname{res}}_{x\in {\left\lvert {X} \right\rvert}} K_x\)
    where \(K_x \subseteq G(F_x)\) is a compact open and almost all
    \(K_x = G({\mathcal{O}}_x)\).
  \item
    \(K^\natural = \prod_{x\in {\left\lvert {X} \right\rvert}} G({\mathcal{O}}_x)\)
  \end{itemize}
\item
  Automorphic functions
  \begin{align*}
  {\mathcal{A}}_K = C^0\qty{\dcoset{G(F)}{G({\mathbb{A}})}{K} \to {\mathbb{C}}}
  .\end{align*}
  Typically \(\dim {\mathcal{A}}_K = \infty\), admits a left action by
  the Hecke algebra
  \begin{align*}
  {\mathcal{H}}_K \coloneqq C_c^0\qty{\dcoset{K}{G({\mathbb{A}})}{K} \to {\mathbb{C}}}
  \end{align*}
  (the compactly supported functions) equipped with convolution with
  unit given by the characteristic function \(\chi_K\) and is defined as
  \begin{align*}
  (f\ast g): {\mathbb{G}}({\mathbb{A}}) &\to {\mathbb{C}}\\
  x &\mapsto \sum_{g\in \dcosetr{ G({\mathbb{A}})}{K} } f(xg) h(g^{-1})
  ,\end{align*}
  where this sum is finite due to the compact support condition.
\item
  \({\mathcal{A}}_{K, c}\) the compactly supported functions in
  \({\mathcal{A}}_K\).
\item
  Cusp forms
  \({\mathcal{A}}_{K, { \mathrm{cusp} }} = \left\{{f\in {\mathcal{A}}_{K, c} {~\mathrel{\Big\vert}~}\dim_{\mathbb{C}}{\mathcal{H}}_K \cdot f < \infty}\right\}\).
\item
  Eigenforms \(f\): \(f\) such that for almost all places \(x\), \(f\)
  is an eigenvector for the action of
  \begin{align*}
  {\mathcal{H}}_{K_x} = C_c(\dcoset{K_x}{G(F_x)}{K_x} \to {\mathbb{C}})
  .\end{align*}
\end{itemize}

The goal of this theory is to study \({\mathcal{A}}_K\) as an object of
\(\mathsf{ {\mathcal{H}}_K}{\hbox{-}}\mathsf{Mod}\).

\end{remark}

\begin{definition}[$\Bun_G$]

Define
\begin{align*}
{\mathsf{Bun}}_G = \left\{{G{\hbox{-}}\text{bundles on } X}\right\} \cong \dcoset{G(F)}{G({\mathbb{A}})}{K^\natural}, \qquad K^\natural \cong \prod_{x\in {\left\lvert {X} \right\rvert}} G({\mathcal{O}}_x)
.\end{align*}

\end{definition}

\begin{example}[?]

For \(G = \operatorname{GL}_n\), passing from a vector bundle to its
frame bundle yields a bijection
\begin{align*}
\left\{{\operatorname{GL}_n{\hbox{-}}\text{bundles}}\right\} &\rightleftharpoons\left\{{\text{Vector bundes of rank } n}\right\} \\
\operatorname{\mathcal{\mathop{\mathrm{Isom}}}}({\mathcal{O}}{ {}^{ \scriptscriptstyle\oplus^{n} }  }, {\mathcal{E}})\curvearrowleft\operatorname{GL}_n &\mapsfrom {\mathcal{E}}
,\end{align*}
where the Isom sheaf is regarded as principal \(G{\hbox{-}}\)bundles
over \(X\). This generalizes the frame bundle construction.

\end{example}

\begin{observation}

Due to Weil: enrich to sets with automorphisms, i.e.~groupoids. Then
there is an equivalence of groupoids
\begin{align*}
\dcoset{\operatorname{GL}_n(F)}{\operatorname{GL}_n({\mathbb{A}})}{K^\natural} &{ { \, \xrightarrow{\sim}\, }}\mathsf{VectBun}_n(X)
.\end{align*}

\end{observation}

\begin{remark}

Let \((g_x) \in \operatorname{GL}_n({\mathbb{A}})\) and assume
\(g_x = 1\) for all \(x\neq x_0\). Assign a lattice in the local field
\begin{align*}
\Lambda_{x_0} \coloneqq g_{x_0} {\mathcal{O}}_{x_0}{ {}^{ \scriptscriptstyle\oplus^{n} }  } \subseteq F_{x_0}{ {}^{ \scriptscriptstyle\oplus^{n} }  }
,\end{align*}
which is an \({\mathcal{O}}_{x_0}\) submodule of rank \(n\). Now
construct a bundle by gluing with the trivial bundle on \(X\) away from
\(x_0\), so glue \(\Lambda_{x_0}\) with
\({\mathcal{O}}_{X\setminus x_0}{ {}^{ \scriptscriptstyle\oplus^{n} } }\)
in the following way: let \(j: X\setminus\left\{{x_0}\right\} \to X\)
and form
\(j_* {\mathcal{O}}_{X\setminus x_0}{ {}^{ \scriptscriptstyle\oplus^{n} } }\),
which is no longer coherent and it quasicoherent, so looks like
meromorphic functions but with no control on the poles. For
\(U \subseteq X\) an affine open, take the functions regular away from
\(x_0\) and constrain its behavior at \(x_0\) and take the sheaf
associated to the following:
\begin{align*}
U\mapsto {{\Gamma}\qty{U\setminus\left\{{ x_0 }\right\}; {\mathcal{O}}_X{ {}^{ \scriptscriptstyle\oplus^{n} }  } } }\cap\Lambda_{x_0} \subseteq F_{x_0}{ {}^{ \scriptscriptstyle\oplus^{n} }  }
.\end{align*}

\includegraphics{figures/2022-03-05_16-11-56.png}

\end{remark}

\begin{example}[?]

For \(t_{x_0}\) a uniformizer, set
\(g_{x_0} = \operatorname{diag}(t_{x_0}, 1, 1, \cdots, 1)\). The
construction above yields the bundle
\({\mathcal{O}}(-x_0) \oplus {\mathcal{O}}{ {}^{ \scriptscriptstyle\oplus^{(n-1)} } }\).

Conversely, starting a vector bundle, you can get a double coset in
\(\dcoset{\operatorname{GL}_n(F)}{\operatorname{GL}_n({\mathbb{A}})}{K^\natural}\):
for \(V\in \mathsf{VectBun}(X)\), there exists a \(U \subseteq X\) with
\({ \left.{{V}} \right|_{{U}} } \cong {\mathcal{O}}_U{ {}^{ \scriptscriptstyle\oplus^{n} } }\).
Take
\(\Lambda_x = { \left.{{V}} \right|_{{\operatorname{Spec}{\mathcal{O}}_x}} } = g_x {\mathcal{O}}_x{ {}^{ \scriptscriptstyle\oplus^{n} } }\).

\end{example}

\begin{exercise}[?]

Check that this gives an equivalence of groupoids.

\end{exercise}

\begin{remark}

This equivalence holds for more general split \(G\). For
\(G= {\mathsf{Sp}}_{2n}\), a \(G{\hbox{-}}\)bundle is the same as a pair
\((V, \omega)\) where \(V\) is a vector space of rank \(2n\) and
\(\omega V\otimes_{{\mathcal{O}}_x} V\to {\mathcal{O}}_x\) is
symplectic.

\end{remark}

\begin{remark}

So far, this is a pointwise story, so we'll geometrize. It's a fact that
\({\mathsf{Bun}}_G\) is a moduli stack, and its \(k{\hbox{-}}\)points
and \(R_{/ {k}}\) points are
\begin{align*}
{\mathsf{Bun}}_G(k) &= \left\{{G{\hbox{-}}\text{bundles on } X}\right\} \\
{\mathsf{Bun}}_G(R) &= \left\{{G{\hbox{-}}\text{bundles on } X\otimes_k R}\right\} 
.\end{align*}
It's a theorem that these moduli functors are representable by Artin
stacks.

\end{remark}

\begin{example}[?]

Take \(X = {\mathbb{P}}^1\), then
\({\mathsf{Bun}}_G(k){_{\scriptstyle / \sim} }\) can be described in
terms of group-theoretic data. \(G{\hbox{-}}\)bundles for
\(G= \operatorname{GL}_n\) are classified by Grothendieck:
\begin{align*}
\mathsf{vectBun}({\mathbb{P}}^1) &\rightleftharpoons\left\{{d_1 \geq d_2 \geq \cdots \geq d_n {~\mathrel{\Big\vert}~}d_i\in {\mathbb{Z}}}\right\} \\
\bigoplus_i {\mathcal{O}}(d_i) &\mapsfrom \left\{{d_i}\right\}
.\end{align*}

In general, fixing \(T\leq G\) a torus and \(W\) the Weyl group yields
\begin{align*}
{\mathsf{Bun}}_{G/{\mathbb{P}}^1}(k){_{\scriptstyle / \sim} }\rightleftharpoons X_*(T)/W
,\end{align*}
i.e.~bundles are parameterized by the cocharacter lattice, modulo the
Weyl group action.

\end{example}

\begin{question}

We can regard \({\mathcal{A}}_{K^\natural}\) as functions on
\({\mathsf{Bun}}_G(k)\), so what is the \({\mathcal{H}}_K\) action?

\end{question}

\begin{example}[?]

Let \(G=\operatorname{GL}_n\), let \(t_x\) be a uniformizer at \(x\) and
take
\begin{align*}
_x \coloneqq\chi_{S}, S = K_x \operatorname{diag}(t_x,1,1\cdots,1) K_x
.\end{align*}
For \(f: {\mathsf{Bun}}_G(k)\to {\mathbb{C}}\), we get the
\textbf{elementary upper modifier of \(f\)}:
\begin{align*}
f\ast h_x: {\mathsf{Bun}}_G(k) &\to {\mathbb{C}}\\
V &\mapsto \sum_{0\to V \hookrightarrow V' \twoheadrightarrow k_x\to 0} f(V')
.\end{align*}
where \(k_x\) is the skyscraper sheaf at \(x\). This is analogous to
summing over elliptic curves that are \(p{\hbox{-}}\)isogenous to a
given curve.

One could alternatively define a Hecke operator defined by
\begin{align*}
h_x = \chi_S, S\coloneqq K_-x \operatorname{diag}(t_x^{\lambda_1}, t_x^{\lambda_2}, \cdots, t_x^{\lambda_n})
,\end{align*}
where \(\mathbf{\lambda}\) is a collection of integers, and
\begin{align*}
(f\ast h_x)(V) = \sum_{\substack{ V\to V' \\ \lambda, x} }
.\end{align*}

\end{example}

\hypertarget{level-structures}{%
\subsection{Level Structures}\label{level-structures}}

\begin{remark}

For interesting automorphic forms, we need to use more general things
than \(K^\natural\) -- many interesting examples come from
\textbf{parahoric} subgroups of \(G(F_x)\). First we define the
\textbf{Iwahori} as a total preimage of a Borel under a reduction:

\begin{center}
\begin{tikzcd}
    {I_x} && {G({\mathcal{O}}_x)} \\
    \\
    {B(k_x)} && {G(k_x)}
    \arrow["{\text{reduction}}", from=1-3, to=3-3]
    \arrow[hook, from=3-1, to=3-3]
    \arrow[hook, from=1-1, to=1-3]
    \arrow[from=1-1, to=3-1]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwyLCJCKGtfeCkiXSxbMiwyLCJHKGtfeCkiXSxbMiwwLCJHKFxcT09feCkiXSxbMCwwLCJJX3giXSxbMiwxLCJcXHRleHR7cmVkdWN0aW9ufSJdLFswLDEsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzMsMiwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMywwXSxbMywxLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=}{Link
to Diagram}
\end{quote}

For \(G=\operatorname{GL}_2\), one gets
\begin{align*}
I_x = \left\{{{
\begin{bmatrix}
  {a} & {b} 
\\
  {c} & {d}
\end{bmatrix}
} {~\mathrel{\Big\vert}~}a,b,c,d\in {\mathcal{O}}_X, c\in {\mathfrak{m}}_x}\right\}
.\end{align*}
Now parahorics are groups that contain the Iwahori, so there is an
analogy:

\begin{longtable}[]{@{}ll@{}}
\toprule
\(G/k\) & \(G/F_x\) \\
\midrule
\endhead
Borel & Iwahori \\
Parabolic & Parahoric \\
\bottomrule
\end{longtable}

\end{remark}

\begin{remark}

For \(G= \operatorname{GL}_n\) and
\(\Lambda_x \subseteq F_x{ {}^{ \scriptscriptstyle\oplus^{n} } }\), we
can consider
\begin{align*}
{\operatorname{Stab}}_{G(F_x)} (\Lambda_x) = \left\{{g\in \operatorname{GL}_n(F_x){~\mathrel{\Big\vert}~}g\Lambda_x = \Lambda_x}\right\}
.\end{align*}
One can ask for simultaneous stabilizers to get parahorics. In fact, all
parahorics occur as stabilizers of chains of lattices where each stage
differs by dividing by a uniformizer.

\end{remark}

\begin{remark}

To visualize these, one needs \textbf{affine Dynkin diagrams} -- these
are generally obtained by adding a new point connected only to the long
root. In \({\mathsf{G}}_2\), the diagram is:

\includegraphics{figures/2022-03-05_16-41-44.png}

Here taking

\begin{itemize}
\tightlist
\item
  \(\emptyset\) yields the Iwahori,
\item
  \(\left\{{ \alpha_1, \alpha_2}\right\}\) yields \(G({\mathcal{O}}_x)\)
\item
  \(\left\{{ \alpha_0, \alpha_1}\right\}\) yields
  \(P\twoheadrightarrow{\operatorname{SL}}_3\)
\item
  \(\left\{{\alpha_0, \alpha_2}\right\}: Q\twoheadrightarrow{\operatorname{SO}}_4 = ({\operatorname{SL}}_2\times {\operatorname{SL}}_2)/\Delta(\pm I)\).
\end{itemize}

\end{remark}

\hypertarget{zhiwei-yun-lecture-2}{%
\section{Zhiwei Yun, Lecture 2}\label{zhiwei-yun-lecture-2}}

\begin{remark}

Today: what is rigidity?

\end{remark}

\hypertarget{automorphic-data}{%
\subsection{Automorphic Data}\label{automorphic-data}}

\begin{definition}[Automorphic data]

Given the following:

\begin{itemize}
\tightlist
\item
  \(k = {\mathbb{F}}_q\)
\item
  \(S \subseteq {\left\lvert {X} \right\rvert}\) a finite set,
  e.g.~\(X= {\mathbb{P}}^1\) and \(S = \left\{{0, \infty}\right\}\) or
  \(S=\left\{{0,1,\infty}\right\}\).
\item
  For each \(x\in S\), a compact open \(K_x \subseteq G(F_x)\)
\item
  For each \(x\in S\), a character:
\end{itemize}

\begin{center}
\begin{tikzcd}
    {K_x} && {\mathbb{C}}^{\times}\\
    \\
    & {L_\chi \text{ finite}}
    \arrow[from=1-1, to=3-2]
    \arrow[from=3-2, to=1-3]
    \arrow["\chi"', from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJLX3giXSxbMiwwLCJcXENDXFx1bml0cyJdLFsxLDIsIkxfXFxjaGkgXFx0ZXh0eyBmaW5pdGV9Il0sWzAsMl0sWzIsMV0sWzAsMSwiXFxjaGkiLDJdXQ==}{Link
to Diagram}
\end{quote}

The pair \((K_S, \chi_S)\) is \textbf{automorphic data}.

\end{definition}

\begin{remark}

Note that \(\chi_S = 1\) recovers \(f\in {\mathcal{A}}_K\) where
\(K = K_S \times \prod_{x\not\in S} G({\mathcal{O}}_x)\).

\end{remark}

\begin{definition}[Typical data]

A map
\begin{align*}
f\in C^0(\dcoset{G(F)} {G({\mathbb{A}})} {\prod_{x\not\in S} G({\mathcal{O}}_x) } \to {\mathbb{C}})
\end{align*}
is \textbf{\((K_S, \chi_X){\hbox{-}}\)typical} iff
\begin{align*}
f(gk_x) = \chi_x(k_x) f(g)\qquad \forall x\in S, k_x\in K_x, g\in G({\mathbb{A}})
.\end{align*}

\end{definition}

\begin{remark}

We want to make \(\dim {\mathcal{A}}_c(K_S,\chi_S) = 1\). In this case,
the Hecke algebra
\({\mathcal{H}}_{K_y} \curvearrowright f\in {\mathcal{A}}_c(K_S, \chi_S)\)
by a character, making \(f\) a Hecke eigenform.

\end{remark}

\hypertarget{examples-of-naive-ridigidy}{%
\subsection{Examples of naive
ridigidy}\label{examples-of-naive-ridigidy}}

\begin{example}[?]

Let

\begin{itemize}
\tightlist
\item
  \(X={\mathbb{P}}^1\),
\item
  \(G = {\operatorname{SL}}_2\),
\item
  \(S=\left\{{0, 1,\infty}\right\}\),
\item
  \(K_x = I_x\) Iwahori for all \(x\in S\), where
  \begin{align*}
  I_x = \left\{{A\coloneqq{
  \begin{bmatrix}
    {a} & {b} 
  \\
    {c} & {d}
  \end{bmatrix}
  } \in \operatorname{Mat}_{2}({\mathcal{O}}_X) {~\mathrel{\Big\vert}~}c\in {\mathfrak{m}}_x}\right\}
  .\end{align*}
\end{itemize}

Then choosing characters \(\chi_x: k^{\times}\to {\mathbb{C}}^{\times}\)
\emph{generically} will imply \(\dim {\mathcal{A}}_c(K_S, \chi_S) = 1\).
Here generic means that \(\prod \chi_i^{\pm 1}\neq 1\). By global
Langlands for \({\operatorname{SL}}_2\), any
\(f\in {\mathcal{A}}_c(K_S, \chi_S)\) will yield a 2-dimensional local
system on \({\mathbb{P}}^1\setminus\left\{{ 0,1,\infty }\right\}\)
ramified at the 3 punctures. These will be solutions to hypergeometric
differential equations.

For \(G=\operatorname{PGL}_2\) (where the example works similarly), for
\(\chi_0, \chi_1 = 1\) and \(\chi_\infty\) quadratic, there is a cover

\begin{center}
\begin{tikzcd}
\left\{{E_t}\right\}: y^2 = x(x-1)(x-t)
  \ar[d] 
\\
{\mathbb{P}}^1\setminus\left\{{ 0,1,\infty }\right\}
\end{tikzcd}
\end{center}

Moreover \(\left\{{H^1(E_t)}\right\}\) will be a rank 2 local system on
this base.

\end{example}

\begin{example}[?]

Let

\begin{itemize}
\tightlist
\item
  \(X = {\mathbb{P}}^1\)
\item
  \(S = \left\{{0, \infty}\right\}\)
\item
  \(K_0 = I_0\), \(\chi_0 =1\)
\item
  \(\tau\) a uniformizer at \(\infty\)
\item
  \(K_\infty = I_\infty^+ = \left\{{{ \begin{bmatrix}  {a} & {b} \\  {c} & {d} \end{bmatrix} } {~\mathrel{\Big\vert}~}a\equiv d\equiv 1 \operatorname{mod}{\mathfrak{m}}_x, b\in {\mathcal{O}}_x, c\in {\mathfrak{m}}_x}\right\}\),
  the pro \(p\) part
\end{itemize}

There is a map \(K_\infty \to k\) where
\({ \begin{bmatrix}  {a} & {b} \\  {c} & {d} \end{bmatrix} } \to b+{c\over \tau}\operatorname{mod}\tau\).
Any character \(k \xrightarrow{\psi} {\mathbb{C}}^{\times}\) can be
extended to
\(\chi_\infty: K_\infty \to k \xrightarrow{\psi} {\mathbb{C}}^{\times}\),
and \(\dim {\mathcal{A}}_c(K_S, \chi_S)\). This yields a
\textbf{Kloosterman local system} on
\({\mathbb{P}}^1\setminus\left\{{ 0,\infty }\right\}\), where
\begin{align*}
\operatorname{Kl}(a) = \sum_{x\in k^{\times}} \psi\qty{x + a\over x}
\end{align*}
recovers the classical Kloosterman sum by taking trace of Frobenius.

\end{example}

\hypertarget{naive-rigidity}{%
\subsection{Naive Rigidity}\label{naive-rigidity}}

\begin{definition}[Rigidity (Naive Definition)]

\begin{align*}
(K_S, \chi_S) \text{ is rigid }\iff \dim {\mathcal{A}}_c(K_S, \chi_S) = 1
.\end{align*}

\end{definition}

\begin{warnings}

If \(\pi_1 G\neq 1\), then \(\pi_0 {\mathsf{Bun}}_G\geq 2\), yielding
multiple components. It's also not clear if this type of dimension bound
will hold after a base change \(k\to k'\).

\end{warnings}

\hypertarget{base-change}{%
\subsection{Base Change}\label{base-change}}

\begin{remark}

For \(k'/k\) finite, write \(X'\coloneqq X\otimes_k k'\) for the base
change. Let \(S\to S'\) be the preimage of \(S\) in \(S'\), and consider
\(k'_x \coloneqq K_x\otimes_k k'\) How can we base change a character?
We need a norm map to fill in the following diagram:

\begin{center}
\begin{tikzcd}
    {K_x\otimes_k k'} && {K_x} && {\mathbb{C}}^{\times}
    \arrow["{\chi_x}", from=1-3, to=1-5]
    \arrow["{\operatorname{Nm}({-})}", dashed, from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJLX3hcXHRlbnNvcl9rIGsnIl0sWzIsMCwiS194Il0sWzQsMCwiXFxDQ1xcdW5pdHMiXSxbMSwyLCJcXGNoaV94Il0sWzAsMSwiXFxOb3JtKFxcd2FpdCkiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=}{Link
to Diagram}
\end{quote}

\end{remark}

\begin{example}[Base-changing a character]

\begin{center}
\begin{tikzcd}
    {I_x} && k^{\times}&& {\mathbb{C}}^{\times}\\
    \\
    {I_x' = I_x(k')} && {(k')^{\times}} && k^{\times}&& {\mathbb{C}}^{\times}
    \arrow["\chi", from=1-3, to=1-5]
    \arrow[from=1-1, to=1-3]
    \arrow["\chi", from=3-5, to=3-7]
    \arrow["{\operatorname{Nm}_{k'/k}({-})}", from=3-3, to=3-5]
    \arrow[from=3-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNyxbMCwwLCJJX3giXSxbMiwwLCJrXFx1bml0cyJdLFs0LDAsIlxcQ0NcXHVuaXRzIl0sWzAsMiwiSV94JyA9IElfeChrJykiXSxbMiwyLCIoaycpXFx1bml0cyJdLFs0LDIsImtcXHVuaXRzIl0sWzYsMiwiXFxDQ1xcdW5pdHMiXSxbMSwyLCJcXGNoaSJdLFswLDFdLFs1LDYsIlxcY2hpIl0sWzQsNSwiXFxOb3JtX3trJy9rfShcXHdhaXQpIl0sWzMsNF1d}{Link
to Diagram}
\end{quote}

Here
\(I_x(k') = \left\{{a,b,c,d\in {\mathcal{O}}_x\widehat{\otimes} k' \cong k'{\left[\left[ t \right]\right] }}\right\}\).

\end{example}

\begin{example}[?]

\begin{center}
\begin{tikzcd}
    {I_x} && k^{\times}&& {\mathbb{C}}^{\times}\\
    \\
    {I_x' = I_x(k')} && {(k')^{\times}} && k^{\times}&& {\mathbb{C}}^{\times}\\
    {{
\begin{bmatrix}
  {a} & {b} 
\\
  {c} & {d}
\end{bmatrix}
} } && {b + {c\over \tau}\operatorname{mod}\tau}
    \arrow["\chi", from=1-3, to=1-5]
    \arrow[from=1-1, to=1-3]
    \arrow["\chi", from=3-5, to=3-7]
    \arrow["{\operatorname{Trace}_{k'/k}({-})}", from=3-3, to=3-5]
    \arrow[from=3-1, to=3-3]
    \arrow[maps to, from=4-1, to=4-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOSxbMCwwLCJJX3giXSxbMiwwLCJrXFx1bml0cyJdLFs0LDAsIlxcQ0NcXHVuaXRzIl0sWzAsMiwiSV94JyA9IElfeChrJykiXSxbMiwyLCIoaycpXFx1bml0cyJdLFs0LDIsImtcXHVuaXRzIl0sWzYsMiwiXFxDQ1xcdW5pdHMiXSxbMCwzLCJcXG1hdHQgYSBiIGMgZCAiXSxbMiwzLCJiICsge2NcXG92ZXIgXFx0YXV9XFxtb2QgXFx0YXUiXSxbMSwyLCJcXGNoaSJdLFswLDFdLFs1LDYsIlxcY2hpIl0sWzQsNSwiXFxUcmFjZV97aycva30oXFx3YWl0KSJdLFszLDRdLFs3LDgsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=}{Link
to Diagram}
\end{quote}

\end{example}

\begin{remark}

We now geometrize this process to send characters to character sheaves,
i.e.~rank one local systems on \(K_x\). We have a way of taking
\((K_S, \chi_S)\) to \((K'_S, \chi_S')\) for extensions \(K\to K'\), so
we can form \({\mathcal{A}}(k'; K'_S, \chi'_S)\).

\end{remark}

\begin{definition}[Weakly rigid]

Automorphic data \((K_S, \chi_S)\) is \textbf{weakly rigid} iff
\(\dim {\mathcal{A}}_c(k'; K'_S, \chi_S')\) is uniformly bounded for all
extensions \(k\to k'\).

\end{definition}

\hypertarget{relevant-points}{%
\subsection{Relevant Points}\label{relevant-points}}

\begin{remark}

Recall that there is a bijection
\begin{align*}
\dcoset{G(F)}{G({\mathbb{A}})}{K} \rightleftharpoons{\mathsf{Bun}}_G(K)(k)
,\end{align*}
so functions \(f\in {\mathcal{A}}_c(K_S, \chi_S)\) are functions on
\({\mathsf{Bun}}_G(K_S^+)(k)\) where for \(x\in S\),
\(K^+_x {~\trianglelefteq~}K_x\) with
\({ \left.{{\chi_x}} \right|_{{K_x^+}} } = 1\) and \(K_x/K_x^+\) are the
\(k{\hbox{-}}\)points of a finite dimensional group \(L_x\).

\end{remark}

\begin{example}[?]

\(I_x^+{~\trianglelefteq~}I_x = K_x \to {\mathbb{G}}_m(k)\).

\end{example}

\begin{example}[?]

\(I_\infty^{++} = K_\infty^+ {~\trianglelefteq~}I_\infty^+ = K_\infty \to k{ {}^{ \scriptscriptstyle\oplus^{2} } }\)
where
\({ \begin{bmatrix}  {a} & {b} \\  {c} & {d} \end{bmatrix} } \mapsto \qty{b, {c\over \tau}} \operatorname{mod}\tau\).

\end{example}

\begin{remark}

There is a right action
\begin{align*}
C^0({\mathsf{Bun}}_G(K_S^+)(k) \to {\mathbb{C}}?) \curvearrowleft\prod_{x\in S} L_x(k)
,\end{align*}
and the eigenfunctions with eigenvalues \((\chi_x)_{x\in S}\) are in
\({\mathcal{A}}_c(K_S, \chi_S)\). As a set, \({\mathsf{Bun}}_G(K_S^+)\)
has commuting left and right actions, where quotienting by the right
action yields a principal homogeneous space. The left action is by
\(\mathop{\mathrm{Aut}}({\mathcal{E}})\) for
\({\mathcal{E}}\in {\mathsf{Bun}}_G(K_S)\), and permutes points in the
fiber in \(\tilde {\mathcal{E}}\in {\mathsf{Bun}}_G(K_S^+)\). So there
is an evaluation map which is well-defined up to conjugacy
\begin{align*}
\operatorname{ev}_{\mathcal{E}}: \mathop{\mathrm{Aut}}({\mathcal{E}}) \to \prod_{x\in S} L_x(k)
.\end{align*}

\end{remark}

\begin{definition}[Relevant points]

A \(k{\hbox{-}}\)point \({\mathcal{E}}\in {\mathsf{Bun}}_G(K_s)(k)\) is
\textbf{\((K_S, \chi_S){\hbox{-}}\)relevant} iff
\begin{align*}
{ \left.{{ \operatorname{ev}_{\mathcal{E}}^*\qty{ \prod_{x\in S} \chi_x } }} \right|_{{\mathop{\mathrm{Aut}}({\mathcal{E}})^0(k) }} } = 1
.\end{align*}
Similarly one can define relevant \(k'{\hbox{-}}\)points for \(k'/k\) a
finite extension.

\end{definition}

\begin{fact}

\begin{align*}
\dim {\mathcal{A}}_c(k'; K_S, \chi_S) \leq {\sharp}\operatorname{Rel}(K_S), \quad \operatorname{Rel}(K_S) \coloneqq\left\{{(K_S', \chi_S'){\hbox{-}}\text{relevant }k' \text{ points of } {\mathsf{Bun}}_G(K_S) }\right\}
.\end{align*}
Note that taking connected components in the definition is needed to
make this stable under base change.

\end{fact}

\begin{corollary}[?]

\((K_S, \chi_S)\) is weakly rigid
\(\iff {\sharp}\operatorname{Rel}(K_S) < \infty\).

\end{corollary}

\begin{example}[?]

Let

\begin{itemize}
\tightlist
\item
  \(G={\operatorname{SL}}_2\)
\item
  \(K_x = I_x\)
\item
  \(x\in S \coloneqq\left\{{0, 1,\infty}\right\}\)
\end{itemize}

Note that
\begin{align*}
{\mathsf{Bun}}_G(K_S)(k) = \left\{{ V\in \mathsf{VectBundle}^{\operatorname{rank}= 2}, \iota: \bigwedge\nolimits^2 V\cong {\mathcal{O}}_X, \left\{{\ell_x \subseteq V_x}\right\}_{x\in S} }\right\}
,\end{align*}
where the \(\ell\) are lines. So these are bundles with extra structure
at fixed places, and are parameterized by 5-tuples
\({\mathcal{E}}= (V, \iota, \ell_0, \ell_1, \ell_\infty)\). For all
\(x\in S\) we have
\begin{align*}
I_x &\to {\mathbb{G}}_m = L_x \\
{
\begin{bmatrix}
  {a} & {b} 
\\
  {c} & {d}
\end{bmatrix}
} &\mapsto a\operatorname{mod}\tau
,\end{align*}
and the evaluation map is
\(\operatorname{ev}: \mathop{\mathrm{Aut}}({\mathcal{E}}) \to \prod_{x\in S} {\mathbb{G}}_m\).
Each \(\gamma\in \mathop{\mathrm{Aut}}({\mathcal{E}})\) is a map
\(V\to V\) where \(\gamma_0 \curvearrowright V_0\) preserving
\(\ell_0\). Is it the case that
\begin{align*}
{ \left.{{\prod_{x\in S} \chi_x}} \right|_{{\mathop{\mathrm{Aut}}({\mathcal{E}})^0}} } \underset{?}{=} 1
.\end{align*}
For \(V = {\mathcal{O}}^2\) and \(\ell_x \subseteq k^2\) in generic
position,
\(\mathop{\mathrm{Aut}}({\mathcal{E}}) = \left\{{\pm 1}\right\}\) so
they are relevant. Other points are irrelevant: if \(V = L \oplus L'\)
with \(\ell_x\in L_x\) or \(L_x'\),
\(\mathop{\mathrm{Aut}}({\mathcal{E}})\) will contain a copy of
\({\mathbb{G}}_m\) that acts by scaling each \(L\) which will map
nontrivially to \(\prod_{x\in S} {\mathbb{G}}_m\). Since
\(\prod \chi_i^{\pm 1}\neq 1\), we get
\({ \left.{{\operatorname{ev}_{\mathcal{E}}^* \prod \chi_x}} \right|_{{{\mathbb{G}}_m}} } \neq 1\).

\end{example}

\hypertarget{zhiwei-yun-lecture-3}{%
\section{Zhiwei Yun, Lecture 3}\label{zhiwei-yun-lecture-3}}

\begin{remark}

The Langlands correspondence:

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.27}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.72}}@{}}
\toprule
Automorphic & Galois \\
\midrule
\endhead
\(G\) & \(G {}^{ \vee }\) \\
Eigenforms \(f\in {\mathcal{A}}_c(K_S, \chi_S)\) & Local systems,
\(\pi_1(X\setminus S) \to G {}^{ \vee }(\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu)\) \\
Rigid automorphic data & Rigid local systems \\
\bottomrule
\end{longtable}

Today we'll discuss going from the automorphic side to the Galois side
by designing rigid automorphic data.

\end{remark}

\hypertarget{numerical-rigidity}{%
\subsection{Numerical Rigidity}\label{numerical-rigidity}}

\begin{remark}

Automorphic functions will be functions on the algebraic stack
\({\mathsf{Bun}}_G(K_S)\), so we want to consider its
\(k{\hbox{-}}\)points. This stack should have (possibly negative)
dimension at most zero, what does this tell us about the level groups?
For a curve \(C\), there is a formula:
\begin{align*}
\dim {\mathsf{Bun}}_G(K_S) = 0 \iff \sum_{x\in S} [G({\mathcal{O}}_x): K_x] = (1-g(C))\dim G
,\end{align*}
where the brackets indicate \textbf{relative dimension}, which is always
non-negative. Recall that \(I_x\) is a preimage of a Borel under
reduction, and for \(K_x = I_x\) we have
\begin{align*}
[G({\mathcal{O}}_x): I_x] = \dim G({\mathcal{O}}_x)/I_x = \dim G/B = {\sharp}\Phi^+
.\end{align*}
If \(K_x\) is not contained in \(G({\mathcal{O}}_x)\), then
\begin{align*}
[G({\mathcal{O}}_x): K_x] = \dim G({\mathcal{O}}_x)/G({\mathcal{O}}_x) \cap K_x - \dim K_x/G({\mathcal{O}}_x) \cap K_x
.\end{align*}

The RHS in the formula is non-negative only when \(g=0, 1\), so we
expect most rigid data to come from \({\mathbb{P}}^1\). Genus 1 is a
very special case, we get \(K_x\sim G({\mathcal{O}}_x)\).

\end{remark}

\begin{example}[?]

Consider

\begin{itemize}
\tightlist
\item
  \(G\) a fixed group,
\item
  \(X = {\mathbb{P}}^1\),
\item
  \(S = \left\{{0,1,\infty}\right\}\),
\item
  \(K_x\) a parahoric.
\end{itemize}

Then
\begin{align*}
\dim G = \sum_{x=0,1,\infty} [G({\mathcal{O}}_x): K_x]
.\end{align*}
If \(K_x\) corresponds to a subdiagram of a Dynkin diagram, we can read
off the reductive quotient \(L_x\) to get a surjective quotient map
\(K_x\twoheadrightarrow L_x\). In this case,
\begin{align*}
[G({\mathcal{O}}_x): K_x] = {1\over 2}\qty{\dim G - \dim L_x}
.\end{align*}
The condition then becomes
\begin{align*}
\dim G = \sum_{x=0,1,\infty}\dim L_x
.\end{align*}

\end{example}

\begin{example}[?]

Consider the same setup for \(G={\mathsf{G}}_2\).

\includegraphics{figures/2022-03-07_11-48-25.png}

For \(G = {\mathbf{E}}_8\), take \(L_1\) to be the Iwahori and:

\includegraphics{figures/2022-03-07_11-53-37.png}

Idea: delete a node to try to get a group of roughly half-dimension.
Cook up an order 2 character on the reductive quotient for \(x=0\):

\begin{center}
\begin{tikzcd}
    {K_0} && {} \\
    \\
    {L_0(k)} \\
    \\
    {L_0(k)/{\operatorname{Spin}}_{16}(k) \cong C_2} && {\mu_2 = \left\{{\pm 1}\right\}}
    \arrow[two heads, from=1-1, to=3-1]
    \arrow[two heads, from=3-1, to=5-1]
    \arrow[from=5-1, to=5-3]
    \arrow["{\chi_0}", dashed, from=1-1, to=5-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJLXzAiXSxbMiwwXSxbMCwyLCJMXzAoaykiXSxbMCw0LCJMXzAoaykvXFxTcGluX3sxNn0oaykgXFxjb25nIENfMiJdLFsyLDQsIlxcbXVfMiA9IFxcdHN7XFxwbSAxfSJdLFswLDIsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFsyLDMsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFszLDRdLFswLDQsIlxcY2hpXzAiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=}{Link
to Diagram}
\end{quote}

Setting \(\chi_\infty = \chi_1 = 1\), this yields automorphic datum
which turns out to be rigid.

\end{example}

\hypertarget{matching-with-local-monodromy}{%
\subsection{Matching with local
monodromy}\label{matching-with-local-monodromy}}

\begin{remark}

Given a local system, restrict to a formal neighborhood of a puncture to
get a representation of the local Galois group, which we can restrict to
inertia:

\begin{center}
\begin{tikzcd}
    {{ \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu_x/F_x)} && {G {}^{ \vee }(\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu)} \\
    \\
    {{\mathrm{In}}_x}
    \arrow[hook, from=3-1, to=1-1]
    \arrow["{\rho_x\in \mathsf{LocSys}}", from=1-1, to=1-3]
    \arrow["{\rho_x}"', dashed, from=3-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXEdhbChcXGJhcntGfV94L0ZfeCkiXSxbMiwwLCJHXFxkdWFsKFxcYmFyXFxRUWxhZGljKSJdLFswLDIsIlxcSW5lcnRpYV94Il0sWzIsMCwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMCwxLCJcXHJob194XFxpbiBcXExvY1xcU3lzIl0sWzIsMSwiXFxyaG9feCIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

\end{remark}

\begin{example}[?]

If \(K_x = I_x\) and one forms the character
\(K_x\to T(k)\to \mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu^{\times}\),
the representation \(\rho_x\) is tamely ramified. There is a canonical
map to the residue field \({\mathrm{In}}_x \to k_x^{\times}\), and
\(\chi\) can be turned into a morphism
\(k_x^{\times}\to T {}^{ \vee }(\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu^{\times})\)
to the dual torus. The composite dual character
\({\mathrm{In}}_x \to k_x^{\times}\to T {}^{ \vee }(\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu^{\times})\)
has finite order and yields the semisimplification
\(({ \left.{{\rho_x}} \right|_{{{\mathrm{In}}_x}} })^{{\mathrm{ss}}}\).
This yields unipotent monodromy, usually ``maximally'' nontrivial.

\end{example}

\begin{example}[?]

For
\(K_x = I_x^+ = \left\{{{ \begin{bmatrix}  {a} & {b} \\  {c} & {d} \end{bmatrix} } {~\mathrel{\Big\vert}~}a-1,d-1,c\equiv 0 \operatorname{mod}t}\right\}\),
one gets a character:

\begin{center}
\begin{tikzcd}
    {K_x=I_x^+} && k && {\mathbb{C}}^{\times}\\
    {{
\begin{bmatrix}
  {a} & {b} 
\\
  {c} & {d}
\end{bmatrix}
}} && {b + {c\over t}\operatorname{mod}t}
    \arrow["\psi", from=1-3, to=1-5]
    \arrow[from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJLX3g9SV94XisiXSxbMiwwLCJrIl0sWzQsMCwiXFxDQ1xcdW5pdHMiXSxbMCwxLCJcXG1hdHQgYSBiIGMgZCJdLFsyLDEsImIgKyB7Y1xcb3ZlciB0fVxcbW9kIHQiXSxbMSwyLCJcXHBzaSJdLFswLDFdXQ==}{Link
to Diagram}
\end{quote}

One gets a wildly ramified representation:
\begin{align*}
\rho_x: { \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu_x/F_x)\to \operatorname{GL}_2(\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu)
.\end{align*}
The Swan conductor is
\({\mathrm{Sw}}(\rho_x) = 1 = {1\over 2} + {1\over 2}\), where the first
factor comes from the shape of the level group \(I_x^+\). There is a
\textbf{Moy-Prasad filtration} on \(I_x\) indexed by
\({1\over h}{\mathbb{Z}}\) for \(h\) the Coxeter number of \(G\), which
for \(G={\operatorname{SL}}_2\) yields \(h=2\). The filtration is

\begin{center}
\begin{tikzcd}
    {I_x = I_x(0)} \\
    { I_x^+ = I_x\qty{1\over 2}} \\
    {I_x(1)} \\
    {I_x\qty{3\over 2}} \\
    \vdots
    \arrow[hook, from=5-1, to=4-1]
    \arrow[hook, from=4-1, to=3-1]
    \arrow[hook, from=3-1, to=2-1]
    \arrow[hook, from=2-1, to=1-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJJX3ggPSBJX3goMCkiXSxbMCwxLCIgSV94XisgPSBJX3hcXHF0eXsxXFxvdmVyIDJ9Il0sWzAsMiwiSV94KDEpIl0sWzAsMywiSV94XFxxdHl7M1xcb3ZlciAyfSJdLFswLDQsIlxcdmRvdHMiXSxbNCwzLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFszLDIsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzIsMSwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMSwwLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

If \(K_x \subseteq P_x(r)\) for \(r\in {\mathbb{Q}}\), then all slopes
of \(\rho_x\) are at most \(r\), bounding the ramification. In this case
depth matches up with slopes.

\end{example}

\begin{remark}

Ansatz for finding correct automorphic data: if \((K_S, \chi_S)\) is
rigid and \(\rho: \pi_1\to G {}^{ \vee }\), there should be an equality
involving \(a\) the Artin conductor:
\begin{align*}
[G({\mathcal{O}}_x): K_x] = {1\over 2} a({ \operatorname{Ad} }\rho_x)
.\end{align*}

\end{remark}

\begin{example}[Epipelagic automorphic data]

Let

\begin{itemize}
\tightlist
\item
  \(S = \left\{{0, \infty}\right\}\),
\item
  \(K_0 = P_0\) a parahoric,
\item
  \(K_\infty = P_\infty^+\) the pro-unipotent of some parahoric
\item
  \(\chi: k\to {\mathbb{C}}^{\times}\) be an additive character
\end{itemize}

Compose to get a character
\(P_\infty^+ \xrightarrow{?} k \xrightarrow{\psi} {\mathbb{C}}^{\times}\),
where the missing morphism is the interesting bit. For
\(G = {\mathsf{Sp}}_{2n} = {\mathsf{Sp}}(V)\), a Siegel parabolic is the
stabilizer of a Lagrangian subspace in \(V\) and has the following
shape:

\includegraphics{figures/2022-03-07_12-14-49.png}

Write this as \(P_{\mathrm{Sieg}}\) preserving a Lagrangian \(L\) and
take its transpose to get \(P_{\mathrm{Sieg}}^{\operatorname{op}}\)
which preserves a complementary Lagrangian subspace
\(L^c \cong L {}^{ \vee }\). Let
\(P_0 \subseteq G({\mathcal{O}}_0), P_\infty \subseteq G({\mathcal{O}}_\infty)\)
be the associated parahorics. Define a map
\begin{align*}
P_\infty^+ = \left\{{{
\begin{bmatrix}
  {A} & {B} 
\\
  {C} & {D}
\end{bmatrix}
} \in G({\mathcal{O}}_\infty) {~\mathrel{\Big\vert}~}A-I, D-I, C\equiv 0 \operatorname{mod}\tau}\right\} 
&\to W = \operatorname{Sym}^2(L) \oplus \operatorname{Sym}^2(L {}^{ \vee }) \\
{
\begin{bmatrix}
  {A} & {B} 
\\
  {C} & {D}
\end{bmatrix}
} &\mapsto \qty{B\operatorname{mod}\tau, {C\over \tau} \operatorname{mod}\tau}
.\end{align*}

We then have

\begin{itemize}
\tightlist
\item
  \(\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu\in \operatorname{GL}(L)\)
\item
  \(\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu\in \operatorname{GL}(L {}^{ \vee })\)
\item
  \(\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu: L {}^{ \vee }\to L\)
  self-dual, so
  \(\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu {}^{ \vee }= \mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu\),
\item
  \(C/\tau: L\to L {}^{ \vee }\) a self-adjoint operator
\end{itemize}

We can further apply the trace pairing, fixing
\(S\in \operatorname{Sym}^2(L)\) and
\(T\in \operatorname{Sym}^2(L {}^{ \vee })\):
\begin{align*}
\operatorname{Trace}: W\to k \\
(X, Y) &\mapsto \operatorname{Trace}(XT) + \operatorname{Trace}(YS)
.\end{align*}
Choosing a pair \((S, T)\) yields a character:
\begin{align*}
P_\infty^+ \to W \xrightarrow{(S, T)} k \xrightarrow{\psi} {\mathbb{C}}^{\times}
.\end{align*}
\textbf{Stable pairs} \((S, T)\) will yield rigid data, where stable is
the open condition that \(ST\in \mathop{\mathrm{End}}(L)\) has distinct
nonzero eigenvalues in
\(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\), so
regular semisimple and invertible.

\end{example}

\begin{remark}

Epipelagic reps of \(G(F_\infty)\) due to Reeder-J-K. Yu: for
\({\mathsf{Sp}}_{2n}\) this amounts to choosing a matrix of the
following shape with equally sized blocks:

\includegraphics{figures/2022-03-07_12-25-29.png}

\end{remark}

\hypertarget{zhiwei-yun-lecture-4}{%
\section{Zhiwei Yun, Lecture 4}\label{zhiwei-yun-lecture-4}}

\hypertarget{kloosterman-automorphic-systems}{%
\subsection{Kloosterman Automorphic
Systems}\label{kloosterman-automorphic-systems}}

\begin{remark}

We've just been on the automorphic side: today we harvest on the Galois
side! For \((K_S, \chi_S)\) automorphic data, there is a Hecke action
\begin{align*}
f\in {\mathcal{A}}_c(K_S,\chi_S) \curvearrowleft{\mathcal{H}}_{K_x},\quad x\in {\left\lvert {X} \right\rvert} \setminus S
.\end{align*}
The Satake isomorphism yields a correspondence
\begin{align*}
\left\{{\text{Functions } {\mathcal{H}}_{K_x} \to \mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu}\right\}
\rightleftharpoons
\left\{{\text{Semisimple conjugacy classes in } \widehat{G}(\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu)}\right\}
.\end{align*}
So for all places \(x\not\in S\), one gets a Satake parameter
\(\sigma_x\in \widehat{G}_{{\mathrm{ss}}}(\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu)\).
By Langlands, there exists a representation
\(\rho: \pi_1(X\setminus S) \to \widehat{G}(\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu)\)
such that \(\rho(\operatorname{Frob}_x)^{{\mathrm{ss}}} \sim \sigma_x\).
How do you construct \(\rho\) from \(f\)?

\end{remark}

\begin{remark}

We'll geometrize this along the lines of Drinfeld, Laumon, etc. Set
\(G=\operatorname{GL}_n\) and let \(T_x\in {\mathcal{H}}_{K_x}\) be the
characteristic function on
\(K_x \operatorname{diag}(t_x, 1,\cdots, 1) K_x\). For
\(f: {\mathsf{Bun}}_G(k)\to \mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu\),
define an operator
\begin{align*}
(T_x f)({\mathcal{E}}) \coloneqq\sum_{{\mathcal{E}}' \hookrightarrow{\mathcal{E}}\text{ length 1 at } x} f({\mathcal{E}}')
.\end{align*}
Note that the index set is isomorphic to
\({\mathbb{P}}({\mathcal{E}}_x)\). This translates functions to sheaves:
summing corresponds to taking cohomology, characters become character
sheaves. Let
\({\mathsf{Hk} }^1 = \left\{{{\mathcal{E}}' \to {\mathcal{E}}\text{ of length 1}}\right\}\),
then there is a span:

\begin{center}
\begin{tikzcd}
    && {{\mathsf{Hk} }^1} \\
    \\
    {{\mathsf{Bun}}_G} &&&& {{\mathsf{Bun}}_G\times (X\setminus S)}
    \arrow["{h_1}"', from=1-3, to=3-1]
    \arrow["{h_2}", from=1-3, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXEJ1bl9HIl0sWzIsMCwiXFxIa14xIl0sWzQsMiwiXFxCdW5fR1xcdGltZXMgKFhcXHNtIFMpIl0sWzEsMCwiaF8xIiwyXSxbMSwyLCJoXzIiXV0=}{Link
to Diagram}
\end{quote}

The operator \(T_x\) geometrizes in the following way:
\begin{align*}
T_1 {\mathcal{F}}\coloneqq(h_2){}_{!}(h_1{}^{*}{\mathcal{F}}) \in {\mathsf{Sh}}({\mathsf{Bun}}_G\times (X\setminus S))
,\end{align*}
and being an \emph{eigensheaf} translates \(T_x f = \lambda_x f\) for
\(\lambda_x\in \mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu\)
to the condition
\begin{align*}
T_1{\mathcal{F}}= {\mathcal{F}}\boxtimes E
.\end{align*}
The goal is to compute \(E\); this will yield
\begin{align*}
\operatorname{Trace}(\operatorname{Frob}_x; E) = \lambda_x \qquad\forall x\in X\setminus S
.\end{align*}

\end{remark}

\begin{example}[Kloosterman automorphic datum]

Benedict Gross constructed \(\operatorname{Kl}\) automorphic datum,
showed rigidity using a trace formula, and conjectured some properties
of \(\rho\) related to a Kloosterman local system. Heinloth-Ngo-Y.
constructed this \(\rho\), uncovering the story of rigidity here.

Let

\begin{itemize}
\tightlist
\item
  \(G = \operatorname{PGL}_n\)
\item
  The 0-level \(K_0 = I_0\) is the Iwahori and \(\chi_0 = 1\)
\item
  The infinity level \(K_\infty = I_\infty^+\), and the character is
  given in the following way: add superdiagonal and lower-left corner
  mod \(\tau\) (the uniformizer at \(\infty\)), so
  \begin{align*}
  (a_{ij}) \mapsto \sum a_{i, i+1} + {a_{n, 1} \over \tau} \operatorname{mod}\tau \leadsto I_{\infty}^+ \to k \xrightarrow{\psi} \mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu^{\times}
  .\end{align*}
\end{itemize}

In this case,
\begin{align*}
{\mathsf{Bun}}_G(K_0, K_\infty) = \left\{{
\begin{array}{l}
V\in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }^{\operatorname{rank}= n}({\mathbb{P}}^1), \\
F^n \supseteq F^{n-1} \supseteq\cdots\supseteq F^1 \text{ a full flag on } V_0, \\
F_0 \subseteq F_1 \subseteq \cdots\subseteq F^n \text{a full flag on }V_\infty, \\
\left\{{e_i}\right\} \text{a basis of }{\mathsf{gr}\,}_i (F_\bullet)
\end{array}
}\right\}/{\operatorname{Pic}}
.\end{align*}
There is a unique relevant point on each component of
\({\mathsf{Bun}}_G(K_0, K_\infty)\), where
\(\deg V \operatorname{mod}n\) is well-defined. It's given by
\({\mathcal{E}}_0\) where
\({\mathcal{O}}{ {}^{ \scriptscriptstyle\oplus^{n} } } = \bigoplus_{i\leq n} {\mathcal{O}}_i\),
with a flag
\(\left\{{e_n}\right\}, \left\{{e_n, e_{n-1}}\right\},\cdots\). One can
show that \(\mathop{\mathrm{Aut}}({\mathcal{E}}_0) = 1\) making it
automatically relevant.

A point \({\mathcal{E}}_1\) yields
\(\bigoplus_{k\leq n-1} {\mathcal{O}}e_k \oplus {\mathcal{O}}(1)e_n\),
with flags

\begin{itemize}
\tightlist
\item
  \(F^\bullet: \left\{{e_{n-1}}\right\}, \left\{{e_{n-1}, e_{n-2}}\right\},\cdots\).
\item
  \(F_\bullet: \left\{{e_1}\right\}, \left\{{e_1, e_2}\right\}, \cdots\)
\end{itemize}

There is a Hecke stack \({\mathsf{Hk} }\) containing
\(\left\{{\phi: {\mathcal{E}}_0 \hookrightarrow{\mathcal{E}}_1}\right\}\),
and a span:

\begin{center}
\begin{tikzcd}
    && {\left\{{\phi: {\mathcal{E}}_0\hookrightarrow{\mathcal{E}}_1}\right\}} \\
    \\
    {{\mathbb{A}}^n\cong I_\infty^+/I\infty^{++} \xrightarrow{\text{sum}}{\mathbb{A}}^1} &&&& {{\mathbb{P}}^1\setminus\left\{{ 0,\infty }\right\}, \mathop{\mathrm{supp}}\operatorname{coker}\phi} \\
    {AS_\psi} &&&& E
    \arrow["{\operatorname{ev}_\infty}"', from=1-3, to=3-1]
    \arrow["\pi", from=1-3, to=3-5]
    \arrow[dashed, from=4-1, to=4-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMiwwLCJcXHRze1xccGhpOiBcXG1jZV8wXFxlbWJlZHNcXG1jZV8xfSJdLFswLDIsIlxcQUFeblxcY29uZyBJX1xcaW5mdHleKy9JXFxpbmZ0eV57Kyt9IFxcbWFwc3ZpYXtcXHRleHR7c3VtfX1cXEFBXjEiXSxbNCwyLCJcXFBQXjFcXHNtdHN7MCxcXGluZnR5fSwgXFxzdXBwIFxcY29rZXJcXHBoaSJdLFswLDMsIkFTX1xccHNpIl0sWzQsMywiRSJdLFswLDEsIlxcZXZfXFxpbmZ0eSIsMl0sWzAsMiwiXFxwaSJdLFszLDQsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

Pull-push yields a local system. Similarly:

\begin{center}
\begin{tikzcd}
    && {{\mathbb{G}}_m{ {}^{ \scriptscriptstyle\times^{n} }  }} \\
    \\
    {{\mathbb{A}}^1} &&&& {{\mathbb{G}}_m}
    \arrow["{\pi = \text{prod}}", from=1-3, to=3-5]
    \arrow["{\sigma = \text{sum}}"', from=1-3, to=3-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXEFBXjEiXSxbMiwwLCJcXEdHX21cXGNhcnRwb3dlcntufSJdLFs0LDIsIlxcR0dfbSJdLFsxLDIsIlxccGkgPSBcXHRleHR7cHJvZH0iXSxbMSwwLCJcXHNpZ21hID0gXFx0ZXh0e3N1bX0iLDJdXQ==}{Link
to Diagram}
\end{quote}

Defining
\(E \coloneqq{\mathbb{R}}^{n-1} \pi{}_{!}\sigma{}^{*}\mathrm{AS}_\psi \in \mathsf{LocSys}^{\operatorname{rank}= n}\)
exactly recovers Deligne's Kloosterman sheaf.

\end{example}

\begin{remark}

For more general \(G\), \({\mathcal{H}}_{K_x}\) has a Kazhdan basis
\(C_\lambda\), where dominant weights \(\lambda \in X_*(T)\) correspond
to irreducible reps of \(G {}^{ \vee }\). Taking \(T_x\) for
\(\operatorname{GL}_n\) recovers the standard representation of
\(G {}^{ \vee }= \operatorname{GL}_n\). The geometric incarnation of the
Hecke operator is \(T_\lambda F\):

\begin{center}
\begin{tikzcd}
    && {{\mathsf{Hk} }_\lambda} \\
    \\
    {{\mathsf{Bun}}_G(K_S)} &&&& {{\mathsf{Bun}}_G(K_S)\times (X\setminus S)} \\
    {\mathcal{F}}&&&& {T_\lambda {\mathcal{F}}}
    \arrow[from=1-3, to=3-1]
    \arrow[from=1-3, to=3-5]
    \arrow["{({-})\otimes\operatorname{\mathcal{IC}}_\lambda}"', dashed, maps to, from=4-1, to=4-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMiwwLCJcXEhrX1xcbGFtYmRhIl0sWzAsMiwiXFxCdW5fRyhLX1MpIl0sWzQsMiwiXFxCdW5fRyhLX1MpXFx0aW1lcyAoWFxcc20gUykiXSxbMCwzLCJcXG1jZiJdLFs0LDMsIlRfXFxsYW1iZGEgXFxtY2YiXSxbMCwxXSxbMCwyXSxbMyw0LCIoXFx3YWl0KVxcdGVuc29yIFxcSUNfXFxsYW1iZGEiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn0sImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

Here \({\mathcal{F}}\) is an eigensheaf, so
\(T_\lambda {\mathcal{F}}= {\mathcal{F}}\boxtimes E_\lambda\). Note that
the \(\operatorname{\mathcal{IC}}\) sheaf is not always constant.

\begin{fact}

\(\lambda \mapsto E_\lambda\) comes from a
\(\widehat{G}{\hbox{-}}\)local system on \(X\setminus S\):

\begin{center}
\begin{tikzcd}
    {\pi_1(X\setminus S)} && {\widehat{G}} \\
    \\
    && {\operatorname{GL}(V_\lambda)}
    \arrow["\rho", from=1-1, to=1-3]
    \arrow[from=1-3, to=3-3]
    \arrow["{E_\lambda}"', from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHBpXzEoWFxcc20gUykiXSxbMiwwLCJcXGhhdCBHIl0sWzIsMiwiXFxHTChWX1xcbGFtYmRhKSJdLFswLDEsIlxccmhvIl0sWzEsMl0sWzAsMiwiRV9cXGxhbWJkYSIsMl1d}{Link
to Diagram}
\end{quote}

\end{fact}

\end{remark}

\hypertarget{applications}{%
\subsection{Applications}\label{applications}}

\begin{remark}

If \((K_S, \chi_S)\) is ``tame'', where \(K_x\) is parahoric, this data
will make sense over any base field \(k\) which \(\chi_S\) is replaced
by a character sheaf. Note that this only works for multiplicative
characters, since additive characters depend on characteristic. One can
construct these Hecke eigensheaves and \(G {}^{ \vee }\) local systems
for arbitrary fields, e.g.~for
\({\mathbb{P}}^1_{/ {{\mathbb{Q}}}} \setminus S\) where there may not
even be a theory of automorphic forms. A first example constructs an
\({\mathbf{E}}_8{\hbox{-}}\)local system on
\({\mathbb{P}}^1_{/ {{\mathbb{Q}}}} \setminus\left\{{ 0,1,\infty }\right\}\),
yielding a motive whose motivic Galois group is \({\mathbf{E}}_8\). One
can then apply this to the inverse Galois problem, arguing that there
exists a number field \(K\) such that
\begin{align*}
{ \mathsf{Gal}} (K/{\mathbb{Q}}) \cong {\mathbf{E}}_8({\mathbb{F}}_\ell), \qquad \ell \gg 0
.\end{align*}

See the notes for relations to ``rigidity methods'' in inverse Galois
theory.

\end{remark}

\hypertarget{open-problems}{%
\subsection{Open Problems}\label{open-problems}}

\begin{question}

Classification: say \(G=\operatorname{GL}_n\), can one classify all
rigid automorphic data?

\end{question}

\begin{remark}

These should correspond under Langlands to rigid local systems, where
there is an algorithmic classification due to Katz in the tame case and
Arinkin in general. One can start with rank 1 local systems and apply
one of three simple procedures to get local systems of higher rank. Note
that hypergeometric local systems occur.

\end{remark}

\begin{question}

Is there an algorithmic way of producing automorphic data?

\end{question}

\begin{question}

Is there a uniform way to check rigidity?

\end{question}

\begin{remark}

Checking rigidity requires knowing the specific geometry of
\({\mathsf{Bun}}_G\) and some tricky linear algebra. There are some
results that provide the uniform bound on dimensions \({\mathcal{A}}_c\)
needed to prove weak rigidity.

\end{remark}

\begin{question}

Can \({\mathcal{A}}_c(K_S, \chi_S)\) be further decomposed into Hecke
modules when the dimension is bigger than 1?

\end{question}

\begin{remark}

This dimension can grow exponentially.

\end{remark}

\hypertarget{saturday-march-05}{%
\section{Saturday, March 05}\label{saturday-march-05}}

Goal: proving rigidity. Start with example 2.1.5 in the notes on
Kloosterman automorphic data.

\addsec{ToDos}
\listoftodos[List of Todos]
\cleardoublepage

% Hook into amsthm environments to list them.
\addsec{Definitions}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={definition}, numwidth=3.5em]
\cleardoublepage

\addsec{Theorems}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={theorem,proposition}, numwidth=3.5em]
\cleardoublepage

\addsec{Exercises}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={exercise}, numwidth=3.5em]
\cleardoublepage

\addsec{Figures}
\listoffigures
\cleardoublepage


\printbibliography[title=Bibliography]


\end{document}
