# Zhiwei Yun: Rigidity method for automorphic forms over function fields (Lecture 1) :::{.remark} Goal: construct automorphic data over function fields and working out the Langlands correspondence for such examples. Setup: - $k= \FF_q$, - $X\slice k$ a projective, smooth, geometrically connected algebraic curve.[^interestingcase] - $F\da k(X)$ its function field, noting that $\trdeg F = 1$. - $\abs{X} \da \Places(X)$, and $\OO_x \surjects k_x$ with $F_x \subseteq \OO_x$, $F_x = k_x\fls{t_x}$ will be formal Laurent series and $\OO_x \cong k\fps{t_x}$. - $G\slice k$ a split semisimple group, e.g. $G=\SL_n, \PGL_n, \Sp_{2n}, \G_2, \E_8,\cdots$ - The adeles $\AA = \resprod_{x\in \abs X} F_x$, where the $F_x$ will be nonarchimedean fields, - $G(\AA) = \resprod_{x\in \abs X} G(F_x)$ where almost all components are in $G(\OO_x)$. - Level groups: - $K = \resprod_{x\in \abs X} K_x$ where $K_x \subseteq G(F_x)$ is a compact open and almost all $K_x = G(\OO_x)$. - $K^\natural = \prod_{x\in \abs X} G(\OO_x)$ - Automorphic functions \[ \mca_K = C^0\qty{\dcoset{G(F)}{G(\AA)}{K} \to \CC} .\] Typically $\dim \mca_K = \infty$, admits a left action by the Hecke algebra \[ \mch_K \da C_c^0\qty{\dcoset{K}{G(\AA)}{K} \to \CC} \] (the compactly supported functions) equipped with convolution with unit given by the characteristic function $\chi_K$ and is defined as \[ (f\convolve g): \GG(\AA) &\to \CC \\ x &\mapsto \sum_{g\in \dcosetr{ G(\AA)}{K} } f(xg) h(g\inv) ,\] where this sum is finite due to the compact support condition. - $\mca_{K, c}$ the compactly supported functions in $\mca_K$. - Cusp forms $\mca_{K, \cusp} = \ts{f\in \mca_{K, c} \st \dim_\CC \mch_K \cdot f < \infty}$. - Eigenforms $f$: $f$ such that for almost all places $x$, $f$ is an eigenvector for the action of \[ \mch_{K_x} = C_c(\dcoset{K_x}{G(F_x)}{K_x} \to \CC) .\] The goal of this theory is to study $\mca_K$ as an object of $\modsleft{ \mch_K}$. [^interestingcase]: This is already interesting in the case of $X = \PP^1$. ::: :::{.definition title="$\Bun_G$"} Define \[ \Bun_G = \ts{G\dash\text{bundles on } X} \cong \dcoset{G(F)}{G(\AA)}{K^\natural}, \qquad K^\natural \cong \prod_{x\in \abs X} G(\OO_x) .\] ::: :::{.example title="?"} For $G = \GL_n$, passing from a vector bundle to its frame bundle yields a bijection \[ \ts{\GL_n\dash\text{bundles}} &\mapstofrom \ts{\text{Vector bundes of rank } n} \\ \sheaf{\Isom}(\OO\sumpower{n}, \mce)\actsonl \GL_n &\mapsfrom \mce ,\] where the Isom sheaf is regarded as principal $G\dash$bundles over $X$. This generalizes the frame bundle construction. ::: :::{.observation} Due to Weil: enrich to sets with automorphisms, i.e. groupoids. Then there is an equivalence of groupoids \[ \dcoset{\GL_n(F)}{\GL_n(\AA)}{K^\natural} &\iso \mathsf{VectBun}_n(X) .\] ::: :::{.remark} Let $(g_x) \in \GL_n(\AA)$ and assume $g_x = 1$ for all $x\neq x_0$. Assign a lattice in the local field \[ \Lambda_{x_0} \da g_{x_0} \OO_{x_0}\sumpower{n} \subseteq F_{x_0}\sumpower{n} ,\] which is an $\OO_{x_0}$ 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 $\OO_{X\sm x_0}\sumpower{n}$ in the following way: let $j: X\sm\ts{x_0} \to X$ and form $j_* \OO_{X\sm x_0}\sumpower{n}$, 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: \[ U\mapsto \globsec{U\smts{x_0}; \OO_X\sumpower{n} }\intersect \Lambda_{x_0} \subseteq F_{x_0}\sumpower{n} .\] ![](figures/2022-03-05_16-11-56.png) ::: :::{.example title="?"} For $t_{x_0}$ a uniformizer, set $g_{x_0} = \diag(t_{x_0}, 1, 1, \cdots, 1)$. The construction above yields the bundle $\OO(-x_0) \oplus \OO\sumpower{(n-1)}$. Conversely, starting a vector bundle, you can get a double coset in $\dcoset{\GL_n(F)}{\GL_n(\AA)}{K^\natural}$: for $V\in \mathsf{VectBun}(X)$, there exists a $U \subseteq X$ with $\ro{V}{U} \cong \OO_U\sumpower{n}$. Take $\Lambda_x = \ro{V}{\spec \OO_x} = g_x \OO_x\sumpower{n}$. ::: :::{.exercise title="?"} Check that this gives an equivalence of groupoids. ::: :::{.remark} This equivalence holds for more general split $G$. For $G= \Sp_{2n}$, a $G\dash$bundle is the same as a pair $(V, \omega)$ where $V$ is a vector space of rank $2n$ and $\omega: V\tensor_{\OO_x} V\to \OO_x$ is symplectic. ::: :::{.remark} So far, this is a pointwise story, so we'll geometrize. It's a fact that $\Bun_G$ is a moduli stack, and its $k\dash$points and $R\slice k$ points are \[ \Bun_G(k) &= \ts{G\dash\text{bundles on } X} \\ \Bun_G(R) &= \ts{G\dash\text{bundles on } X\tensor_k R} .\] It's a theorem that these moduli functors are representable by Artin stacks. ::: :::{.example title="?"} Take $X = \PP^1$, then $\Bun_G(k)\modiso$ can be described in terms of group-theoretic data. $G\dash$bundles for $G= \GL_n$ are classified by Grothendieck: \[ \mathsf{vectBun}(\PP^1) &\mapstofrom \ts{d_1 \geq d_2 \geq \cdots \geq d_n \st d_i\in \ZZ} \\ \bigoplus_i \OO(d_i) &\mapsfrom \ts{d_i} .\] In general, fixing $T\leq G$ a torus and $W$ the Weyl group yields \[ \Bun_{G/\PP^1}(k)\modiso \mapstofrom X_*(T)/W ,\] i.e. bundles are parameterized by the cocharacter lattice, modulo the Weyl group action. ::: :::{.question} We can regard $\mca_{K^\natural}$ as functions on $\Bun_G(k)$, so what is the $\mch_K$ action? ::: :::{.example title="?"} Let $G=\GL_n$, let $t_x$ be a uniformizer at $x$ and take \[ _x \da \chi_{S}, S = K_x \diag(t_x,1,1\cdots,1) K_x .\] For $f: \Bun_G(k)\to \CC$, we get the **elementary upper modifier of $f$**: \[ f\convolve h_x: \Bun_G(k) &\to \CC \\ V &\mapsto \sum_{0\to V \injects V' \surjects k_x\to 0} f(V') .\] where $k_x$ is the skyscraper sheaf at $x$. This is analogous to summing over elliptic curves that are $p\dash$isogenous to a given curve. One could alternatively define a Hecke operator defined by \[ h_x = \chi_S, S\da K_-x \diag(t_x^{\lambda_1}, t_x^{\lambda_2}, \cdots, t_x^{\lambda_n}) ,\] where $\vector \lambda$ is a collection of integers, and \[ (f\convolve h_x)(V) = \sum_{\substack{ V\birational V' \\ \lambda, x} } .\] ::: ## Level Structures :::{.remark} 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: \begin{tikzcd} {I_x} && {G(\OO_x)} \\ \\ {B(k_x)} && {G(k_x)} \arrow["{\text{reduction}}", from=1-3, to=3-3] \arrow[hook, from=3-1, to=3-3] \arrow[hook, from=1-1, to=1-3] \arrow[from=1-1, to=3-1] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJCKGtfeCkiXSxbMiwyLCJHKGtfeCkiXSxbMiwwLCJHKFxcT09feCkiXSxbMCwwLCJJX3giXSxbMiwxLCJcXHRleHR7cmVkdWN0aW9ufSJdLFswLDEsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzMsMiwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMywwXSxbMywxLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=) For $G=\GL_2$, one gets \[ I_x = \ts{\matt a b c d \st a,b,c,d\in \OO_X, c\in \mfm_x} .\] Now parahorics are groups that contain the Iwahori, so there is an analogy: | $G/k$ | $G/F_x$ | |----------- |----------- | | Borel | Iwahori | | Parabolic | Parahoric | | | | ::: :::{.remark} For $G= \GL_n$ and $\Lambda_x \subseteq F_x\sumpower{n}$, we can consider \[ \Stab_{G(F_x)} (\Lambda_x) = \ts{g\in \GL_n(F_x)\st g\Lambda_x = \Lambda_x} .\] 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. ::: :::{.remark} To visualize these, one needs **affine Dynkin diagrams** -- these are generally obtained by adding a new point connected only to the long root. In $\G_2$, the diagram is: ![](figures/2022-03-05_16-41-44.png) Here taking - $\emptyset$ yields the Iwahori, - $\ts{ \alpha_1, \alpha_2}$ yields $G(\OO_x)$ - $\ts{ \alpha_0, \alpha_1}$ yields $P\surjects \SL_3$ - $\ts{\alpha_0, \alpha_2}: Q\surjects \SO_4 = (\SL_2\times \SL_2)/\Delta(\pm I)$. :::