#  Zhiwei Yun, Lecture 4

## Kloosterman Automorphic Systems

:::{.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
\[
f\in \mca_c(K_S,\chi_S) \actsonl \mch_{K_x},\quad x\in \abs{X} \sm S
.\]
The Satake isomorphism yields a correspondence
\[
\ts{\text{Functions } \mch_{K_x} \to \bar{\QQladic}}
\mapstofrom
\ts{\text{Semisimple conjugacy classes in } \hat{G}(\bar{\QQladic})}
.\]
So for all places $x\not\in S$, one gets a Satake parameter $\sigma_x\in \hat{G}_{\semisimple}(\bar{\QQladic})$.
By Langlands, there exists a representation $\rho: \pi_1(X\sm S) \to \hat G(\bar{\QQladic})$ such that $\rho(\Frob_x)^{\semisimple} \sim \sigma_x$.
How do you construct $\rho$ from $f$?
:::

:::{.remark}
We'll geometrize this along the lines of Drinfeld, Laumon, etc.
Set $G=\GL_n$ and let $T_x\in \mch_{K_x}$ be the characteristic function on $K_x \diag(t_x, 1,\cdots, 1) K_x$.
For $f: \Bun_G(k)\to \bar{\QQladic}$, define an operator 
\[
(T_x f)(\mce) \da \sum_{\mce' \embeds \mce \text{ length 1 at } x} f(\mce')
.\]
Note that the index set is isomorphic to $\PP(\mce_x)$.
This translates functions to sheaves: summing corresponds to taking cohomology, characters become character sheaves.
Let $\Hk^1 = \ts{\mce' \to \mce \text{ of length 1}}$, then there is a span:

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

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXEJ1bl9HIl0sWzIsMCwiXFxIa14xIl0sWzQsMiwiXFxCdW5fR1xcdGltZXMgKFhcXHNtIFMpIl0sWzEsMCwiaF8xIiwyXSxbMSwyLCJoXzIiXV0=)

The operator $T_x$ geometrizes in the following way:
\[
T_1 \mcf \da (h_2)\lshriek (h_1\pushf \mcf) \in \Sh(\Bun_G\times (X\sm S))
,\]
and being an *eigensheaf* translates $T_x f = \lambda_x f$ for $\lambda_x\in \bar{\QQladic}$ to the condition
\[
T_1\mcf = \mcf \boxtensor E
.\]
The goal is to compute $E$; this will yield
\[
\Trace(\Frob_x; E) = \lambda_x \qquad\forall x\in X\sm S
.\]
:::

:::{.example title="Kloosterman automorphic datum"}
Benedict Gross constructed $\Kl$ 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 

- $G = \PGL_n$
- The 0-level $K_0 = I_0$ is the Iwahori and $\chi_0 = 1$
- 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
\[
(a_{ij}) \mapsto \sum a_{i, i+1} + {a_{n, 1} \over \tau} \mod\tau \leadsto I_{\infty}^+ \to k \mapsvia{\psi} \bar{\QQladic}\units
.\]

In this case, 
\[
\Bun_G(K_0, K_\infty) = \ts{
\begin{array}{l}
V\in \VectBundle^{\rank = n}(\PP^1), \\
F^n \contains F^{n-1} \contains \cdots\contains F^1 \text{ a full flag on } V_0, \\
F_0 \subseteq F_1 \subseteq \cdots\subseteq F^n \text{a full flag on }V_\infty, \\
\ts{e_i} \text{a basis of }\gr_i (F_\bullet)
\end{array}
}/\Pic
.\]
There is a unique relevant point on each component of $\Bun_G(K_0, K_\infty)$, where $\deg V \mod n$ is well-defined.
It's given by $\mce_0$ where $\OO\sumpower n = \bigoplus_{i\leq n} \OO_i$, with a flag $\ts{e_n}, \ts{e_n, e_{n-1}},\cdots$.
One can show that $\Aut(\mce_0) = 1$ making it automatically relevant.

A point $\mce_1$ yields \( \bigoplus_{k\leq n-1} \OO e_k \oplus \OO(1)e_n \), with flags 

- $F^\bullet: \ts{e_{n-1}}, \ts{e_{n-1}, e_{n-2}},\cdots$.
- $F_\bullet: \ts{e_1}, \ts{e_1, e_2}, \cdots$

There is a Hecke stack $\Hk$ containing $\ts{\phi: \mce_0 \embeds \mce_1}$, and a span:

\begin{tikzcd}
	&& {\ts{\phi: \mce_0\embeds\mce_1}} \\
	\\
	{\AA^n\cong I_\infty^+/I\infty^{++} \mapsvia{\text{sum}}\AA^1} &&&& {\PP^1\smts{0,\infty}, \supp \coker\phi} \\
	{AS_\psi} &&&& E
	\arrow["{\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}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCJcXHRze1xccGhpOiBcXG1jZV8wXFxlbWJlZHNcXG1jZV8xfSJdLFswLDIsIlxcQUFeblxcY29uZyBJX1xcaW5mdHleKy9JXFxpbmZ0eV57Kyt9IFxcbWFwc3ZpYXtcXHRleHR7c3VtfX1cXEFBXjEiXSxbNCwyLCJcXFBQXjFcXHNtdHN7MCxcXGluZnR5fSwgXFxzdXBwIFxcY29rZXJcXHBoaSJdLFswLDMsIkFTX1xccHNpIl0sWzQsMywiRSJdLFswLDEsIlxcZXZfXFxpbmZ0eSIsMl0sWzAsMiwiXFxwaSJdLFszLDQsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

Pull-push yields a local system.
Similarly:

\begin{tikzcd}
	&& {\GG_m\cartpower{n}} \\
	\\
	{\AA^1} &&&& {\GG_m}
	\arrow["{\pi = \text{prod}}", from=1-3, to=3-5]
	\arrow["{\sigma = \text{sum}}"', from=1-3, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXEFBXjEiXSxbMiwwLCJcXEdHX21cXGNhcnRwb3dlcntufSJdLFs0LDIsIlxcR0dfbSJdLFsxLDIsIlxccGkgPSBcXHRleHR7cHJvZH0iXSxbMSwwLCJcXHNpZ21hID0gXFx0ZXh0e3N1bX0iLDJdXQ==)

Defining $E \da \RR^{n-1} \pi\lshriek \sigma\pushf \mathrm{AS}_\psi \in \Locsys^{\rank = n}$ exactly recovers Deligne's Kloosterman sheaf. 
:::

:::{.remark}
For more general $G$, $\mch_{K_x}$ has a Kazhdan basis $C_\lambda$, where dominant weights $\lambda \in X_*(T)$ correspond to irreducible reps of $G\dual$.
Taking $T_x$ for $\GL_n$ recovers the standard representation of $G\dual = \GL_n$.
The geometric incarnation of the Hecke operator is $T_\lambda F$:

\begin{tikzcd}
	&& {\Hk_\lambda} \\
	\\
	{\Bun_G(K_S)} &&&& {\Bun_G(K_S)\times (X\sm S)} \\
	\mcf &&&& {T_\lambda \mcf}
	\arrow[from=1-3, to=3-1]
	\arrow[from=1-3, to=3-5]
	\arrow["{(\wait)\tensor \IC_\lambda}"', dashed, maps to, from=4-1, to=4-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCJcXEhrX1xcbGFtYmRhIl0sWzAsMiwiXFxCdW5fRyhLX1MpIl0sWzQsMiwiXFxCdW5fRyhLX1MpXFx0aW1lcyAoWFxcc20gUykiXSxbMCwzLCJcXG1jZiJdLFs0LDMsIlRfXFxsYW1iZGEgXFxtY2YiXSxbMCwxXSxbMCwyXSxbMyw0LCIoXFx3YWl0KVxcdGVuc29yIFxcSUNfXFxsYW1iZGEiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn0sImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

Here $\mcf$ is an eigensheaf, so $T_\lambda \mcf = \mcf\boxtensor E_\lambda$.
Note that the $\IC$ sheaf is not always constant.

:::{.fact}
$\lambda \mapsto E_\lambda$ comes from a $\hat G\dash$local system on $X\sm S$:

\begin{tikzcd}
	{\pi_1(X\sm S)} && {\hat G} \\
	\\
	&& {\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}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHBpXzEoWFxcc20gUykiXSxbMiwwLCJcXGhhdCBHIl0sWzIsMiwiXFxHTChWX1xcbGFtYmRhKSJdLFswLDEsIlxccmhvIl0sWzEsMl0sWzAsMiwiRV9cXGxhbWJkYSIsMl1d)

:::

:::

## Applications

:::{.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 $G\dual$ local systems for arbitrary fields, e.g. for $\PP^1\slice \QQ\sm S$ where there may not even be a theory of automorphic forms.
A first example constructs an $\E_8\dash$local system on $\PP^1\slice \QQ\smts{0,1,\infty}$, yielding a motive whose motivic Galois group is $\E_8$.
One can then apply this to the inverse Galois problem, arguing that there exists a number field $K$ such that 
\[
\Gal(K/\QQ) \cong \E_8(\FF_\ell), \qquad \ell \gg 0
.\]

See the notes for relations to "rigidity methods" in inverse Galois theory.
:::

## Open Problems

:::{.question}
Classification: say $G=\GL_n$, can one classify all rigid automorphic data?
:::

:::{.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.

:::

:::{.question}
Is there an algorithmic way of producing automorphic data? 
:::

:::{.question}
Is there a uniform way to check rigidity?
:::

:::{.remark}
Checking rigidity requires knowing the specific geometry of $\Bun_G$ and some tricky linear algebra.
There are some results that provide the uniform bound on dimensions $\mca_c$ needed to prove weak rigidity.

:::

:::{.question}
Can $\mca_c(K_S, \chi_S)$ be further decomposed into Hecke modules when the dimension is bigger than 1?
:::

:::{.remark}
This dimension can grow exponentially.
:::