A fairy tale: see Langlands Elephant. The point of this talk is to see the entire elephant!

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

• 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.
• 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.
  • Artin/Grothendieck: for finite coefficients, realized this using étale cohomology \(H^*_\text{ét}(X)\).
• 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.

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

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

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) \mathrel{\vcenter{:}}= H^i_\text{ét}(\operatorname{Spec}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }; M) .\end{align*} There is a LES involving \(M {}^{ \vee }\mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{\mathsf{Grp}}(M, S^1) = \mathop{\mathrm{Hom}}_{\mathsf{Grp}}(M, \mu_{p^\infty})\) where \(\mu_{p^\infty}\) are the \(p{\hbox{-}}\)power roots of 1:

Link to Diagram

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

Link to Diagram

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

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

• 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 \({\mathbb{F}}_p\).
• The 2-dimensional objects: \({ {\mathbb{Q}}_p }, {\mathbb{F}}_p{\left(\left( t \right)\right) }\), or projective smooth curves over \(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\).

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

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

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

A heavily studied piece of the analogy: \begin{align*} {\operatorname{ {\text{Automorphic forms}} }}\rightleftharpoons{\mathsf{TQFT}} ,\end{align*} coming from the 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*} .

See Atiyah TQFT, section 2.

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:

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

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

• 4-manifolds \(\leadsto {\mathbb{C}}\)
• 3-manifolds \(\leadsto { \mathsf{Vect} }_{/ {{\mathbb{C}}}}\)
• 2-manifolds \(\leadsto\) categories enriched over \({ \mathsf{Vect} }_{/ {{\mathbb{C}}}}\)

This should yield

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

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)

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

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

The 2-dimensional objects:

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

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!

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 \(p\).

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 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 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!

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?

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 boundary conditions in \(A^{(G)}\) and \(B^{(G {}^{ \vee })}\).

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

• A variety \(G{\hbox{-}}\)variety gives a boundary condition for both \(A^{(G)}\) and \(B^{(G {}^{ \vee })}\),
• For suitable choice of \(Y\), this recovers familiar invariants of automorphic forms mentioned above,
• On the Galois side this recovers \(L{\hbox{-}}\)functions,
• 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.

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

Overall plan:

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

Today:

• Motivations from modular forms,
• Fundamental definitions.

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

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 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 \({\mathsf{G_K}{\hbox{-}}\mathsf{Mod}}\).

Conjectures

• Meanings of \(L{\hbox{-}}\)function values, e.g. Deligne’s conjecture that they come from motives.
• Langlands: connections between Galois reps \(\rho\) and automorphic reps.

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 \(V_R \mathrel{\vcenter{:}}= V\otimes_{K^+} R\) for any \(R\in\mathsf{Alg}_{/ {K^+}}\).

If \(R={\mathbb{R}}\), choose an ordered basis for \(B\) to define the 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) \mathrel{\vcenter{:}}=(a, b) .\end{align*}

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

On the modular form side:

1. \(f: {\mathfrak{h}}\to {\mathbb{C}}, f(z) = (cz+d)^{-k}f(\gamma z)\), holomorphic at cusps, etc
2. \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) \mathrel{\vcenter{:}}= \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
3. Adelic interpretation: \(\operatorname{GL}_2({\mathbb{A}}) = \operatorname{GL}_2({\mathbb{Q}}) \operatorname{GL}_2^+({\mathbb{R}}) \tilde K\) where \(\tilde K \mathrel{\vcenter{:}}=\prod_p K_p\)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.

• Can recover \(\Gamma \mathrel{\vcenter{:}}=\operatorname{GL}_2({\mathbb{Q}}) \cap(\operatorname{GL}_2^+({\mathbb{R}}) \times {\mathcal{K}})\)
• 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*}
• Get functions \begin{align*} \phi_f: \dcosetl{\operatorname{GL}_2({\mathbb{Q}})}{\operatorname{GL}_2({\mathbb{A}})} \to {\mathbb{C}} .\end{align*}

On the automorphic side:

1. 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.
2. Writing \(G\mathrel{\vcenter{:}}=\operatorname{GU}(n, m)\) for a form of signature \((n,m)\), replace with \(\dcosetl{\Gamma Z(G)}{G}\) where \(G\supseteq{\mathcal{K}}_\infty \mathrel{\vcenter{:}}= U(n){ {}^{ \scriptscriptstyle\times^{2} } }\), and analogously \(\dcosetl{\Gamma Z(G)}{G({\mathbb{R}})} \to {\mathbb{C}}\).
3. For \(G({\mathbb{A}}_f) = {\textstyle\coprod}_i G({\mathbb{R}})^{-1}? {\mathcal{K}}\).

An 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 \mathrel{\vcenter{:}}=\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*}

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.

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.

Several perspectives on automorphic forms on unitary groups:

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

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

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

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.

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

• A polarization,
• An endomorphism, and
• A level structure.

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 \({\operatorname{U}}_{a, b}\).

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 \(\omega \mathrel{\vcenter{:}}=\pi_* \Omega_{{\mathcal{A}}/{\mathcal{M}}}\); 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 \({\mathfrak{h}}_n\).

Goal for today: introduce an approach to studying certain \(L{\hbox{-}}\)functions using the 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.

A general strategy:

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

  • Factors into an Euler product
  • Has a functional equation
  • Can be meromorphically continued to \({\mathbb{C}}\)

• 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.

• Express a familiar automorphic \(L{\hbox{-}}\)function in terms of this pairing.

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.

Setup:

• \(K\in \mathsf{Field}_{/ {{\mathbb{Q}}}}\) a quadratic extension
• \(V\in { \mathsf{Vect} }_{/ {K}} ^{\dim = n}\) with a nondegenerate Hermitian pairing \({\left\langle {{-}},~{{-}} \right\rangle}_V\)
• \(G = {\operatorname{U}}(V, {\left\langle {{-}},~{{-}} \right\rangle}_V)\)
• \(W = V{ {}^{ \scriptscriptstyle\oplus^{2} } }\) with the induced Hermitian pairing \begin{align*} {\left\langle {(u, v)},~{(u', v')} \right\rangle}_W \mathrel{\vcenter{:}}={\left\langle {u},~{u'} \right\rangle}_V - {\left\langle {v},~{v'} \right\rangle}_V .\end{align*}
• A doubled group \(H \mathrel{\vcenter{:}}={\operatorname{U}}(W, {\left\langle {{-}},~{{-}} \right\rangle}_W)\)
• 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 \(\operatorname{sig}(a, b) \times \operatorname{sig}(b, a) \to \operatorname{sig}(a+b, a+b)\)

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

Some references on doubling:

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

We’ll continue with the previous setup. Let \(P \leq H\) be parabolic preserving \(V^\Delta \mathrel{\vcenter{:}}=\left\{{(v,v) {~\mathrel{\Big\vert}~}v\in V}\right\}\). 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*}

Setup:

• \(\pi\) a cuspidal automorphic representation of \(G\)
• \(\tilde \pi\) the contragradient (dual) representation of \(\pi\)
• \(\phi\in \pi\)
• \(\tilde\phi\in \tilde\pi\)

Define a zeta integral \begin{align*} Z(\phi, \tilde\phi, f_{s, \chi}) \mathrel{\vcenter{:}}=\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.

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 finite sum.

Shimura computed coefficients in many cases.

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

• Fix \(X\mathrel{\vcenter{:}}=\dcosetl{P}{H}\) and write \(\gamma\in X\), identifying it with its coset \(H\gamma\).
• For each \(\gamma\in X\), write \([G\times G]^\gamma \mathrel{\vcenter{:}}={\operatorname{Stab}}_{G\times G}(\gamma)\).
• Write \([\gamma]\) for the orbit of \(P\gamma\) under the \(G\times G\) action.

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 }\\ &\mathrel{\vcenter{:}}= \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) \mathrel{\vcenter{:}}= \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:

• For \(\gamma = 1\), \(I(\gamma)\) will be the RHS in the theorem statement.
• For \(\gamma\neq 1\), \(I(\gamma) = 0\).

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\} \mathrel{\vcenter{:}}= 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) \mathrel{\vcenter{:}}=(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.

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 \(n=1\)?

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

• \(n=1\)

• \(K\in \mathsf{Field}_{/ {{\mathbb{Q}}}}\) be an imaginary quadratic field

• \(V\in { \mathsf{Vect} }^{\dim = 1}_{/ {K}}\)

• \(W = V{ {}^{ \scriptscriptstyle\oplus^{2} } }\)

• \(G\mathrel{\vcenter{:}}={\operatorname{U}}(V, {\left\langle {{-}},~{{-}} \right\rangle}_V) \cong {\operatorname{U}}_1 = \left\{{g\in \operatorname{GL}_2 {~\mathrel{\Big\vert}~}g\mkern 1.5mu\overline{\mkern-1.5mug\mkern-1.5mu}\mkern 1.5mu = 1}\right\}\). Note that \(G \subseteq \operatorname{GU}(V, {\left\langle {{-}},~{{-}} \right\rangle}_V)\cong \operatorname{GU}_1 \cong \operatorname{GL}_1\)

• \(H\mathrel{\vcenter{:}}={\operatorname{U}}(W, {\left\langle {{-}},~{{-}} \right\rangle}_V) \cong {\operatorname{U}}_{1, 1}\). Note that \(H \subseteq \operatorname{GU}(W, {\left\langle {{-}},~{{-}} \right\rangle}_V)\cong \operatorname{GU}_{1, 1}\)

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.

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.

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:

• \(Z\) is an automorphic form on \(\operatorname{GU}_1 = \operatorname{GL}_1\), and thus a Hecke character.
• 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*}

There is a diagram:

Link to Diagram

Moreover there is an embedding:

Link to Diagram

These induce embeddings of corresponding Shimura varieties:

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

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.

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

• Integral is a finite sum,
• There are only characters,
• We’re evaluating at special points!

Let \(Z(s, \chi) \mathrel{\vcenter{:}}= Z(s, \chi, \phi,\tilde\phi)\). 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 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.

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:

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 \(s=0\)!

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 Gauss-Manin connection and the Kodaira morphism, and a splitting \begin{align*} H^1_{\mathrm{dR}} = \omega \oplus H^{0, 1} \end{align*} which preserves algebraicity at CM points.

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

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\mathrel{\vcenter{:}}= 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*}

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.

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 \({\mathbb{A}}\mathrel{\vcenter{:}}=\prod^{\operatorname{res}}k_v\) 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 \([G] \curvearrowleft G({\mathbb{A}})\).

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.

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*}

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) \mathrel{\vcenter{:}}=\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

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

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

• \(\pi_v\in {\mathsf{Irr}}(G(k_v))\),
• \(\pi_v^{k_v} \neq 0\) for almost all \(v\), so \(k_v\) is unramified or spherical.

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 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 \(\chi\).

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) \mathrel{\vcenter{:}}=\prod_{v\not\in S} L(s, \pi_v, R),\qquad L(s, \pi_v, R) \mathrel{\vcenter{:}}={1\over \operatorname{det}1-q_v^{-s} R( S_{\pi_v} ) }, \quad q\mathrel{\vcenter{:}}=? .\end{align*} These generalize Hecke \(L{\hbox{-}}\)functions and those attached to modular forms.

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 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.

Note that tempered reps are 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.

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 \(G = {\operatorname{U}}_3\).

There are some invariants:

• \(n=\dim V\),
• \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.

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

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:

• \(V\) Hermitian, \(W\) skew-Hermitian
• An invariant \({\operatorname{disc}}(V) \mathrel{\vcenter{:}}=(-1)^m \operatorname{det}V\in F^{\times}/\operatorname{Nm}(E^{\times})\) where \(m\mathrel{\vcenter{:}}={n\choose 2}\) and \(n\mathrel{\vcenter{:}}=\dim V\)
• \({\operatorname{disc}}(W) = {\operatorname{disc}}( \delta^{-n} V)\) where \(\delta\in E_0^{\times}\).

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 coherent.

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({-}, {-}) \mathrel{\vcenter{:}}=\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 theta functions. Note that \(ZU(W) = E^1 \mathrel{\vcenter{:}}=\left\{{x\in E^{\times}{~\mathrel{\Big\vert}~}\operatorname{Nm}(x) = 1}\right\}\). 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 \(E^1\).

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*}

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 \\ &\mathrel{\vcenter{:}}=\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*}

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.

The theta correspondence is an instance of all of these ideas.

Let

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

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 metaplectic cover \({\operatorname{Mp}}\).

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

Link to Diagram

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

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.

One needs a lift of the following form:

Link to Diagram

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

• \({ \left.{{ \chi_V}} \right|_{{F^{\times}}} } = \omega_{E/F}^{\dim V}\)
• \({ \left.{{\chi_W}} \right|_{{F^{\times}}} } = \omega_{E/F}^{\dim W}\)

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

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

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*}

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.

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:

Link to Diagram

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

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

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

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) &\mathrel{\vcenter{:}}=\int_X K(x, y) f(x) \,dx ,\end{align*} so \(K\) acts like a matrix. In our case, we’ll want a lift

Link to Diagram

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

Link to Diagram

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) \mathrel{\vcenter{:}}=\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 global theta lift of \(\pi\) as \begin{align*} \Theta(\pi) \mathrel{\vcenter{:}}=\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*}

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 \(r=0\)?

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*}

Setup:

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

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 \(W' = W_0' \oplus {\mathbb{H}}\).

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 \(L_{F_v} \hookrightarrow L_F\).

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

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

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

From \(\psi\) we’ll obtain

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

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*}

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*}

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.

Plans for lectures:

1. What is \({\mathsf{G}}_2\), what are modular forms on it?
2. Fourier expansions of modular forms on \({\mathsf{G}}_2\).
3. Examples and theorems about modular forms on \({\mathsf{G}}_2\).
4. Beyond \({\mathsf{G}}_2\), possibly \({\mathbf{E}}_8\).

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) &\mathrel{\vcenter{:}}= j(g, z)^{-\ell} f(gz) \\ \\ j\qty{ g = { \begin{bmatrix} {a} & {b} \\ {c} & {d} \end{bmatrix} },z} &\mathrel{\vcenter{:}}= cz+d .\end{align*}

Some properties:

1. Growth: \(\phi_f\) is of moderate growth.
2. Invariance: \(\phi_f(\gamma g) = \phi_f(g)\) for all \(\gamma\in \Gamma \leq {\operatorname{SL}}_2({\mathbb{Z}})\).
3. Equivariance on a compact: \(k_\theta \mathrel{\vcenter{:}}={ \begin{bmatrix} {\cos \theta} & {-\sin \theta} \\ {\sin \theta} & {\cos \theta} \end{bmatrix} }\in {\operatorname{SO}}_2({\mathbb{R}})\) satisfies \(\phi_f(g k_\theta) = e^{-i\ell \theta} \phi_f(g)\)
4. 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 \(D_{{\mathrm{CR}}} \phi_f \mathrel{\vcenter{:}}= X_- f\).

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.

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:

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

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

Todos:

• What is \({\mathsf{G}}_2\)?
• What is \(D_\ell\)?
• What are some examples/theorems about modular forms on \({\mathsf{G}}_2\)?

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'] &\mathrel{\vcenter{:}}=\phi\phi' - \phi'\phi & \phi,\phi'\in {\mathfrak{sl}}_3 \\ [\phi, v] &\mathrel{\vcenter{:}}=\phi(v), & v\in V_3\\ [\phi, \delta] &\mathrel{\vcenter{:}}=\phi(\delta),&\delta\in V_3 {}^{ \vee } .\end{align*}

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

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:

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

This yields a root diagram:

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,

• On \({\mathfrak{sl}}_3\), this acts as \(X\mapsto {}^{-t}X\)
• On \(V_3\), it’s \(V_3\mapsto V_3 {}^{ \vee }\) by \(v_j\mapsto \delta_j\).

Define

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

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

• \({\mathsf{G}}_2\leadsto {\operatorname{SL}}_2\)
• ?

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.

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:

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

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.

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

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

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

Some facts:

• \(Z = \exp({\mathbb{R}}E_{13})\)
• \(W = {\mathbb{R}}E_{12} + {\mathbb{R}}v_1 + {\mathbb{R}}\delta_3 + {\mathbb{R}}E_{23}\)
• \(N/Z = \exp(W)\)
• \(M\curvearrowright Z\) by the determinant
• \(M\curvearrowright N/Z\) as \(\operatorname{Sym}^3(V_3) \otimes\operatorname{det}(V_3)^{-1}\)
• There is a symplectic form on \(W\) where \([w, w'] = {\left\langle {w},~{w'} \right\rangle} E_{13}\)
• 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 \({\left\langle {mw},~{mw'} \right\rangle} = \operatorname{det}(m) {\left\langle {w},~{w'} \right\rangle}\).

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

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

Define \begin{align*} \phi_w(g) \mathrel{\vcenter{:}}=\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.

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*}

Identify \(W\) as a space \(B\) of binary cubics under \begin{align*} W &\to B \\ w \mathrel{\vcenter{:}}= a E_{12} + {b\over 3} v_1 + {c\over 3} \delta_3 + d E_{23} &\mapsto f_w\mathrel{\vcenter{:}}= 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) \mathrel{\vcenter{:}}={\left\langle {w},~{m\cdot (u-? v)^3} \right\rangle} ,\end{align*} which will appear in Fourier expansions.

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

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

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

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 \(E_\ell(g) \mathrel{\vcenter{:}}= E_\ell(g, f, s=\ell+1)\).

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

• \(f_\pi\in \operatorname{Ind}_{P({\mathbb{A}})}^{{\mathsf{G}}_2({\mathbb{A}})}(\pi)\)
• \(E(g, f_\pi) = \sum_{ \gamma\in \dcosetl{P({\mathbb{Q}})}{{\mathsf{G}}_2({\mathbb{Q}})} } f_\pi( \gamma g)\)
• If \(\ell \geq 6\) this is a weight \(\ell\) modular form on \({\mathsf{G}}_2\).
• 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 \(\Delta\).

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

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

There is a theory of 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

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

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.

• \(Q_R\) can be empty
• 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*}

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.

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 \(n_C(xy) = n_C(x) n_C(y)\).

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

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

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 \({\mathfrak{g}}_{J_C}\).

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

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.

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

• \(k= {\mathbb{F}}_q\),

• \(X_{/ {k}}\) a projective, smooth, geometrically connected algebraic curve.¹

• \(F\mathrel{\vcenter{:}}= k(X)\) its function field, noting that \(\operatorname{trdeg}F = 1\).

• \({\left\lvert {X} \right\rvert} \mathrel{\vcenter{:}}={\operatorname{Places}}(X)\), 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 \({\mathcal{O}}_x \cong k{\left[\left[ t_x \right]\right] }\).

• \(G_{/ {k}}\) a split semisimple group, e.g. \(G={\operatorname{SL}}_n, \operatorname{PGL}_n, {\mathsf{Sp}}_{2n}, {\mathsf{G}}_2, {\mathbf{E}}_8,\cdots\)

• The adeles \({\mathbb{A}}= \prod^{\operatorname{res}}_{x\in {\left\lvert {X} \right\rvert}} F_x\), where the \(F_x\) will be nonarchimedean fields,

• \(G({\mathbb{A}}) = \prod^{\operatorname{res}}_{x\in {\left\lvert {X} \right\rvert}} G(F_x)\) where almost all components are in \(G({\mathcal{O}}_x)\).

• Level groups:

  • \(K = \prod^{\operatorname{res}}_{x\in {\left\lvert {X} \right\rvert}} K_x\) where \(K_x \subseteq G(F_x)\) is a compact open and almost all \(K_x = G({\mathcal{O}}_x)\).
  • \(K^\natural = \prod_{x\in {\left\lvert {X} \right\rvert}} G({\mathcal{O}}_x)\)

• 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 \mathrel{\vcenter{:}}= 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.

• \({\mathcal{A}}_{K, c}\) the compactly supported functions in \({\mathcal{A}}_K\).

• Cusp forms \({\mathcal{A}}_{K, { \mathrm{cusp} }} = \left\{{f\in {\mathcal{A}}_{K, c} {~\mathrel{\Big\vert}~}\dim_{\mathbb{C}}{\mathcal{H}}_K \cdot f < \infty}\right\}\).

• 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*}

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

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} \mathrel{\vcenter{:}}= 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*}

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.

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.

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

Link to Diagram

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:

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.

To visualize these, one needs 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:

Here taking

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

Today: what is rigidity?

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

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.

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

Link to Diagram

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 \({\mathcal{A}}(k'; K'_S, \chi'_S)\).

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 \(L_x\).

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*}

The Langlands correspondence:

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

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 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 \(K_x\sim G({\mathcal{O}}_x)\).

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:

Link to Diagram

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*}

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:

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 \(f\)?

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}}) \mathrel{\vcenter{:}}=\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:

Link to Diagram

The operator \(T_x\) geometrizes in the following way: \begin{align*} T_1 {\mathcal{F}}\mathrel{\vcenter{:}}=(h_2){}_{!}(h_1{}^{*}{\mathcal{F}}) \in {\mathsf{Sh}}({\mathsf{Bun}}_G\times (X\setminus S)) ,\end{align*} and being an 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*}

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 \(T_\lambda F\):

Link to Diagram

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.

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.

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.

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.

This dimension can grow exponentially.