# Zhiwei Yun, Lecture 3 :::{.remark} The Langlands correspondence: | Automorphic | Galois | |--------------------------------------- |--------------------------------------------------------- | | $G$ | $G\dual$ | | Eigenforms $f\in \mca_c(K_S, \chi_S)$ | Local systems, $\pi_1(X\sm S) \to G\dual(\bar\QQladic)$ | | Rigid automorphic data | Rigid local systems | Today we'll discuss going from the automorphic side to the Galois side by designing rigid automorphic data. ::: ## Numerical Rigidity :::{.remark} Automorphic functions will be functions on the algebraic stack $\Bun_G(K_S)$, so we want to consider its $k\dash$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: \[ \dim \Bun_G(K_S) = 0 \iff \sum_{x\in S} [G(\OO_x): K_x] = (1-g(C))\dim G ,\] 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 \[ [G(\OO_x): I_x] = \dim G(\OO_x)/I_x = \dim G/B = \size \Phi^+ .\] If $K_x$ is not contained in $G(\OO_x)$, then \[ [G(\OO_x): K_x] = \dim G(\OO_x)/G(\OO_x) \intersect K_x - \dim K_x/G(\OO_x) \intersect K_x .\] The RHS in the formula is non-negative only when $g=0, 1$, so we expect most rigid data to come from $\PP^1$. Genus 1 is a very special case, we get $K_x\sim G(\OO_x)$. ::: :::{.example title="?"} Consider - $G$ a fixed group, - $X = \PP^1$, - $S = \ts{0,1,\infty}$, - $K_x$ a parahoric. Then \[ \dim G = \sum_{x=0,1,\infty} [G(\OO_x): K_x] .\] If $K_x$ corresponds to a subdiagram of a Dynkin diagram, we can read off the reductive quotient $L_x$ to get a surjective quotient map $K_x\surjects L_x$. In this case, \[ [G(\OO_x): K_x] = {1\over 2}\qty{\dim G - \dim L_x} .\] The condition then becomes \[ \dim G = \sum_{x=0,1,\infty}\dim L_x .\] ::: :::{.example title="?"} Consider the same setup for $G=\G_2$. ![](figures/2022-03-07_11-48-25.png) For $G = \E_8$, take $L_1$ to be the Iwahori and: ![](figures/2022-03-07_11-53-37.png) Idea: delete a node to try to get a group of roughly half-dimension. Cook up an order 2 character on the reductive quotient for $x=0$: \begin{tikzcd} {K_0} && {} \\ \\ {L_0(k)} \\ \\ {L_0(k)/\Spin_{16}(k) \cong C_2} && {\mu_2 = \ts{\pm 1}} \arrow[two heads, from=1-1, to=3-1] \arrow[two heads, from=3-1, to=5-1] \arrow[from=5-1, to=5-3] \arrow["{\chi_0}", dashed, from=1-1, to=5-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJLXzAiXSxbMiwwXSxbMCwyLCJMXzAoaykiXSxbMCw0LCJMXzAoaykvXFxTcGluX3sxNn0oaykgXFxjb25nIENfMiJdLFsyLDQsIlxcbXVfMiA9IFxcdHN7XFxwbSAxfSJdLFswLDIsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFsyLDMsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFszLDRdLFswLDQsIlxcY2hpXzAiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=) Setting $\chi_\infty = \chi_1 = 1$, this yields automorphic datum which turns out to be rigid. ::: ## Matching with local monodromy :::{.remark} Given a local system, restrict to a formal neighborhood of a puncture to get a representation of the local Galois group, which we can restrict to inertia: \begin{tikzcd} {\Gal(\bar{F}_x/F_x)} && {G\dual(\bar\QQladic)} \\ \\ {\Inertia_x} \arrow[hook, from=3-1, to=1-1] \arrow["{\rho_x\in \Locsys}", from=1-1, to=1-3] \arrow["{\rho_x}"', dashed, from=3-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXEdhbChcXGJhcntGfV94L0ZfeCkiXSxbMiwwLCJHXFxkdWFsKFxcYmFyXFxRUWxhZGljKSJdLFswLDIsIlxcSW5lcnRpYV94Il0sWzIsMCwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMCwxLCJcXHJob194XFxpbiBcXExvY1xcU3lzIl0sWzIsMSwiXFxyaG9feCIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==) ::: :::{.example title="?"} If $K_x = I_x$ and one forms the character $K_x\to T(k)\to \bar\QQladic\units$, the representation $\rho_x$ is tamely ramified. There is a canonical map to the residue field $\Inertia_x \to k_x\units$, and $\chi$ can be turned into a morphism $k_x\units\to T\dual(\bar\QQladic\units)$ to the dual torus. The composite dual character $\Inertia_x \to k_x\units\to T\dual(\bar\QQladic\units)$ has finite order and yields the semisimplification $(\ro{\rho_x}{\Inertia_x})^{\semisimple}$. This yields unipotent monodromy, usually "maximally" nontrivial. ::: :::{.example title="?"} For $K_x = I_x^+ = \ts{\matt a b c d \st a-1,d-1,c\equiv 0 \mod t}$, one gets a character: \begin{tikzcd} {K_x=I_x^+} && k && \CC\units \\ {\matt a b c d} && {b + {c\over t}\mod t} \arrow["\psi", from=1-3, to=1-5] \arrow[from=1-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJLX3g9SV94XisiXSxbMiwwLCJrIl0sWzQsMCwiXFxDQ1xcdW5pdHMiXSxbMCwxLCJcXG1hdHQgYSBiIGMgZCJdLFsyLDEsImIgKyB7Y1xcb3ZlciB0fVxcbW9kIHQiXSxbMSwyLCJcXHBzaSJdLFswLDFdXQ==) One gets a wildly ramified representation: \[ \rho_x: \Gal(\bar F_x/F_x)\to \GL_2(\bar\QQladic) .\] The Swan conductor is $\Sw(\rho_x) = 1 = {1\over 2} + {1\over 2}$, where the first factor comes from the shape of the level group $I_x^+$. There is a **Moy-Prasad filtration** on $I_x$ indexed by ${1\over h}\ZZ$ for $h$ the Coxeter number of $G$, which for $G=\SL_2$ yields $h=2$. The filtration is \begin{tikzcd} {I_x = I_x(0)} \\ { I_x^+ = I_x\qty{1\over 2}} \\ {I_x(1)} \\ {I_x\qty{3\over 2}} \\ \vdots \arrow[hook, from=5-1, to=4-1] \arrow[hook, from=4-1, to=3-1] \arrow[hook, from=3-1, to=2-1] \arrow[hook, from=2-1, to=1-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJJX3ggPSBJX3goMCkiXSxbMCwxLCIgSV94XisgPSBJX3hcXHF0eXsxXFxvdmVyIDJ9Il0sWzAsMiwiSV94KDEpIl0sWzAsMywiSV94XFxxdHl7M1xcb3ZlciAyfSJdLFswLDQsIlxcdmRvdHMiXSxbNCwzLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFszLDIsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzIsMSwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMSwwLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dXQ==) If $K_x \subseteq P_x(r)$ for $r\in \QQ$, then all slopes of $\rho_x$ are at most $r$, bounding the ramification. In this case depth matches up with slopes. ::: :::{.remark} Ansatz for finding correct automorphic data: if $(K_S, \chi_S)$ is rigid and $\rho: \pi_1\to G\dual$, there should be an equality involving $a$ the Artin conductor: \[ [G(\OO_x): K_x] = {1\over 2} a(\Ad \rho_x) .\] ::: :::{.example title="Epipelagic automorphic data"} Let - $S = \ts{0, \infty}$, - $K_0 = P_0$ a parahoric, - $K_\infty = P_\infty^+$ the pro-unipotent of some parahoric - $\chi: k\to \CC\units$ be an additive character Compose to get a character $P_\infty^+ \mapsvia{?} k \mapsvia{\psi} \CC\units$, where the missing morphism is the interesting bit. For $G = \Sp_{2n} = \Sp(V)$, a Siegel parabolic is the stabilizer of a Lagrangian subspace in $V$ and has the following shape: ![](figures/2022-03-07_12-14-49.png) Write this as $P_\Sieg$ preserving a Lagrangian $L$ and take its transpose to get $P_\Sieg\op$ which preserves a complementary Lagrangian subspace $L^c \cong L\dual$. Let $P_0 \subseteq G(\OO_0), P_\infty \subseteq G(\OO_\infty)$ be the associated parahorics. Define a map \[ P_\infty^+ = \ts{\matt A B C D \in G(\OO_\infty) \st A-I, D-I, C\equiv 0 \mod \tau} &\to W = \Sym^2(L) \oplus \Sym^2(L\dual) \\ \matt A B C D &\mapsto \qty{B\mod \tau, {C\over \tau} \mod \tau} .\] We then have - $\bar A\in \GL(L)$ - $\bar D\in \GL(L\dual)$ - $\bar B: L\dual\to L$ self-dual, so $\bar{B}\dual = \bar B$, - $C/\tau: L\to L\dual$ a self-adjoint operator We can further apply the trace pairing, fixing $S\in \Sym^2(L)$ and $T\in \Sym^2(L\dual)$: \[ \Trace: W\to k \\ (X, Y) &\mapsto \Trace(XT) + \Trace(YS) .\] Choosing a pair $(S, T)$ yields a character: \[ P_\infty^+ \to W \mapsvia{(S, T)} k \mapsvia{\psi} \CC\units .\] **Stable pairs** $(S, T)$ will yield rigid data, where stable is the open condition that $ST\in \Endo(L)$ has distinct nonzero eigenvalues in $\bar k$, so regular semisimple and invertible. ::: :::{.remark} Epipelagic reps of $G(F_\infty)$ due to Reeder-J-K. Yu: for $\Sp_{2n}$ this amounts to choosing a matrix of the following shape with equally sized blocks: ![](figures/2022-03-07_12-25-29.png) :::