# Wee Teck Gan: Automorphic forms and the theta correspondence (Talk 1) :::{.remark} Goal: reformulating the Ramanujan-Petersson conjecture in terms of representation theory. ::: ## The Ramanujan-Petersson conjecture :::{.remark} Let $f: \mfh\to \CC$ 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 \[ f(z) = \sum_{k\geq 1} a_k(f) q^n, \qquad q\da e^{2\pi i z} ,\] which can be normalized so that $a_1(f) = 1$. The remaining coefficients are then the Hecke eigenvalues, so \[ T_p f = a_p(f) f .\] ::: :::{.conjecture title="Ramanujan-Petersson"} \[ \abs{a_p(f) } \leq 2p^{k-1\over 2} .\] This was proved by Deligne as a consequence of the Weil conjectures. There is an analog for **Maass forms**, which involves the **hyperbolic Laplacian**, which similarly bounds Fourier coefficients. ::: :::{.remark} Error terms come from the cusp forms here. There is a bridge that takes holomorphic modular forms and Maass forms to the world of automorphic forms. ::: :::{.remark} Setup: let $k\in \NF, v \in \Places(K)$ so that $k_v\in\Loc\Field$. Define the adeles as $\AA \da \resprod k_v$ which admits a diagonal embedding $k\embeds \AA$ with $\dcosetl{k}{\AA}$ compact. Let $G\in\Alg\Grp\slice k$ be reductive, e.g. $\SL_n, \U_n$, we then similarly have \[ G(k) \embeds G(\AA) = \resprod_v G(k_v) \] with $\ts{k_v}$ an open compact subgroup. Write $[G] = \dcosetl{G(k)}{G(\AA)}$, and note the there is a right action $[G] \actsonl G(\AA)$. ::: ## Automorphic Reps :::{.definition title="Automorphic forms on reductive groups"} An **automorphic form** on $G$ is a function $f: [G]\to \CC$ satisfying - Regularity conditions: e.g. at worst polynomial growth $f\sim z^k$, smoothness, and derivatives $f^{(n)}\sim z^k$ for the same exponent. - Finiteness conditions: $K\dash$finiteness for $K = \prod_v k_v$, or more generally $Z(\lieg)\dash$finiteness. Write $\mca(G)$ for the vector space of automorphic forms on $G$. Note that this carries at left $G(\AA)$ action: \[ (g_0 . f)(g) = f(g g_0) .\] ::: :::{.remark} The finiteness condition will guarantee that $f$ will come from the kernel of a differential operator, e.g. the CR equations for holomorphy. Requiring $K\dash$finiteness only gives an action on finite adeles. ::: :::{.definition title="Automorphic representation"} An **automorphic representation** is an irreducible representation of $\mca(A)$. ::: ## Cusp Forms :::{.definition title="Cusp forms"} A form $f\in \mca(G)$ is **cuspidal** iff for all parabolic subgroups $P$ with $P=MN$, the constant term of $f$ along $N$ is zero, where the constant term is defined as \[ f_N(g) = \int_{[N]} = f(ng) \dn .\] This yields a subspace of cusp forms $\mca_\cusp(G) \leq \mca(G)$ which is stable under the $G(\AA)$ action. ::: :::{.remark} One can take a character $\psi: [N]\to \CC\units$, then there is a $(N, \psi)\dash$Fourier coefficient of $f$: \[ f_{N, \psi}(g) = \int_{[N]} \bar{\psi(n)} \cdot f(ng) \dn .\] ::: :::{.remark} Uniform moderate growth and being cuspidal imply that $f\in \mca_\cusp(G)$ rapidly decays at $\infty$, i.e. faster than $1/p$ for any polynomial, so that $f\in L^2$: \[ \int_{[G]} \abs{f}^2 < \infty .\] So define the Hilbert space of square-integrable automorphic forms \[ \mca_2(G) \da \ts{f\in \mca(G) \st \norm{f}_{L^2}<\infty } .\] There is a containment \[ \mca_\cusp(G) \subseteq \mca_2(G) \subseteq \mca(G) ,\] where there are a decomposition into irreducible reps - $\mca_\cusp(G) = \bigoplus_\pi m(\pi) \pi$ for some cuspidal multiplicities $m$, - $\mca_2(G) = \bigoplus_\pi m(\pi)\pi$ for some $L^2$ multiplicities $m$. ::: :::{.question} A main question for automorphic representations: for which $\pi$ is $m(\pi) > 0$? I.e. which representations occur as cuspidal or $L^2$ reps? Moreover, what do all of the irreducible reps of $G(\AA)$ look like? ::: :::{.remark} Recall that since $G(\AA) = \resprod_v G(k_v)$, we expect a representation $\pi$ of $G(\AA)$ to break up as $\pi = \restensor_v \pi _v$ with - $\pi_v\in \Irr(G(k_v))$, - $\pi_v^{k_v} \neq 0$ for almost all $v$, so $k_v$ is **unramified** or **spherical**. ::: ## Unramified Reps :::{.remark} There is a containment $G_v \contains K_v$ 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 $G_v \contains B_v\contains T_v N_v$, and there is a bijection \[ \Irr\Rep(G_v)(K_v\dash\text{unramified}) &\mapstofrom \ts{\text{Unramified characters of }T_v}/W \\ I(\chi) = \Ind_{B_v}^{G_v} \chi &\mapsfrom \chi ,\] where we mod out by a Weyl group action $W$. Note that $I(\chi)$ is the unique unramified subquotient. There is a further correspondence \[ \ts{\text{Unramified characters of }T_v}/W \\ &\underset{ {\text{Langlands}} }{\mapstofrom} \ts{\text{Semisimple conjugacy classes in } G\dual(\CC)} \\ \chi &\mapsto S_\chi ,\] so there is some semisimple conjugacy class associated to characters $\chi$. ::: :::{.remark} Thus for $\pi\in \mca_\cusp(G)$ with $\pi = \bigotimes^{\res}_v \pi_v$, one gets a collection $\ts{S_{\pi_v} \st v\not\in S} \subseteq G\dual$. For $R: G\dual \to \GL_N(\CC)$, we can form an \(L\dash \)function \[ L^S(s, \pi R) \da \prod_{v\not\in S} L(s, \pi_v, R),\qquad L(s, \pi_v, R) \da {1\over \det 1-q_v^{-s} R( S_{\pi_v} ) }, \quad q\da ? .\] These generalize Hecke \(L\dash \)functions and those attached to modular forms. ::: ## Tempered Reps :::{.remark} A character of the torus $\chi: T_v\to \CC\units$ yields $\pi_\chi$ a $K_v\dash$unramified irrep. Say $\pi_\chi$ is **tempered** iff $\chi$ is unitary, i.e. it factors as $\chi: T_v\to S^1$ so that $\abs{\chi} = 1$. Tempered reps naturally occur as regular representations. ::: :::{.remark} Note that tempered reps are *weakly* contained in $L^2(G_v)$, but not e.g. the trivial representation of $\SL_2$ is not in $L_2(\RR)$, but $\SL_2(\RR)$ does not have finite volume. In general, the trivial representation is not tempered unless the group is compact. ::: :::{.conjecture title="Ramanujan-Petersson, reformulated but false"} Let $\pi = \restensor_v \subseteq \mca_\cusp(G)$ for $G$ quasi-split (or split), then $\pi_v$ is tempered for almost all $v$. ::: :::{.remark} This conjecture is false! There is a counterexample for $G = \SP_4$, and a goal for this course is to construct a counterexample for $G = \U_3$. ::: :::{.conjecture title="Ramanujan-Petersson, reformulated and fixed"} If $\pi \subseteq \mca_\cusp(G)$ and $\pi$ is **globally generic** (a certain big enough Fourier coefficient), then $\pi_v$ is tempered for almost all $v$. ::: ## Unitary groups :::{.definition title="Unitary Groups"} Let - $E/F$ be a quadratic extension, - $\Gal(E/F) = \gens{c}$ is cyclic, - $V\in \Vect\slice E$, - $\inp{\wait}{\wait}: V\cartpower{2}\to E$ which is $\eps\dash$Hermitian for $\eps = \pm 1$, i.e. \[ \inp{av}{bw} = a\inp{v}{w} b^c, \qquad \inp{v}{w} = \eps \inp{w}{v} .\] - \[ \delta \in E_0\units \da \ts{x\in E\units \st \Tr(x) = 0} .\] - $\delta\cdot\inp{\wait}{\wait}$ is $(-\eps)\dash$Hermitian. Then define the **unitary group** as \[ \U(V) \da \Aut(V, \inp{\wait}{\wait}) .\] ::: :::{.remark} There are some invariants: - $n=\dim V$, - \[ \disc(V) = (-1)^{n\choose 2} \det(V)\in F\units/ \Norm(E\units) ,\] 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. :::