--- created: 2022-10-28T16:49 updated: 2023-05-03T15:02 title: automorphic form aliases: [automorphic forms, automorphic, automorphy, double coset] --- - Tags - #arithmetic-geometry/Langlands - Refs: - - Introduction to the [plectic conjecture](Unsorted/plectic%20conjecture.md) #resources/notes - Notes on Fourier analysis on number fields: #resources/course-notes - Links: - [modular form](Unsorted/modular%20form.md) - [automorphic representation](Unsorted/automorphic%20representation.md) - [Shimura variety](Unsorted/Shimura%20variety.md) - [Hecke stack](Hecke%20stack.md) - [[Shimura–Taniyama–Weil conjecture]] - [Fermat's Last Theorem](Unsorted/Fermat's%20Last%20Theorem.md) - [Petersson inner product](Petersson%20inner%20product) --- # automorphic form Slogan: 1. Automorphy 2. $K\dash$finiteness 3. Highly structured ODE 4. Moderate growth. More precisely, ![[2023-03-28-3.png]] ![[2023-03-28-4.png]] New vs classical: ![](2023-05-03.png) - Rewriting the half-plane as a [homogeneous space](Unsorted/homogeneous%20space.md): let $\SL_2(\RR)\actson \HH$ on the right by $z\actsonl \matt abcd\da {ax + b\over cx+d}$, then the stabilizer at $i$ is $\Stab_{\SL_2(\RR)}(i) = \ts{\matt a b {-b} a} = \SO_2(\RR)$. So $$ \HH \cong \dcoset{1}{\SL_2(\RR)}{\Stab_{\SL_2(\RR)}(i) } = \dcoset{1}{\SL_2(\RR)}{\SO_2(\RR) } $$ - Note that $\SO_2(\RR) \injects \SL_2(\RR)$ is the maximal compact subgroup. - Write $\HH = \dcoset{\SL_2(\ZZ)}{\SL_2(\RR)}{1}$ or $\dcoset{1}{\SL_2(\RR)}{\SO_2(\QQ)}$, so $$ X = \dcoset{\SL_2(\ZZ)}{\SL_2(\RR)}{\SO_2(\QQ)} $$ - More generally, replace $\dcoset{\Gamma}{\HH}{1}$ with $\dcoset{\QQ}{\AA}{1}$, i.e. $$ X = \dcoset{\GL_n(\QQ)}{\GL_n(\AA)}{1} $$ - Not sure how this relates yet: a moduli space of unimodular lattices is $$ X = \dcoset{\SL_2(\hat\ZZ)}{\SL_2(\AA_\QQ)}{\SL_2(\QQ)} \cong \dcoset{1}{\SL_2(\RR)}{\SL_2(\ZZ)} $$ - Write the space of automorphic functions as $$ \mca \da C^\infty\qty{\dcoset{\GL_n(\hat\ZZ)}{\GL_n(\AA_{\QQ})}{\GL_n(\QQ) )} \too \CC} $$ - Let $K = \FF_q[C]$ be the function field associated to $C$ a curve, so $\ff(K) = \FF_q(C)$. - Another common setup: can realize certain affine [modular curves](Unsorted/modular%20curve.md) $\dcoset{\Gamma}{\HH}{1}$ as $\dcoset{\GL_2(\QQ)}{\GL_2(\AA_\QQ)}{K}$ where $K = K^\infty \times K_\infty$ and $K^\infty \leq \GL_2(\AA_\QQ^\infty)$ is a compact open and $K_\infty \da \RR_{>0}\times \SO_2(\RR)$. - $\GL_2(\AA_\QQ) \leq \GL_2(\RR) \times \prod'_{p\in \abs{\spec \ZZ}} \GL_2(\QQpadic)$ where almost every entry is in $\GL_2(\ZZpadic)$. An **automorphic form** is a function on $\dcoset{\GL_2(\QQ)}{\GL_2(\AA)}{1}$, instead of things like $\dcoset{\Gamma}{X}{1}$ where $X = \HH, \SL_2(\RR), \GL_2(\RR)$, etc. - $\QQ \injects \AA_\QQ$ is discrete, as is $\GL_2(\QQ)\injects \GL_2(\AA)$. ![](attachments/Pasted%20image%2020220427000823.png) # As line bundles See [Baily-Borel](Unsorted/Baily-Borel.md). ![](attachments/Pasted%20image%2020230314200531.png) # Double coset spaces ![](attachments/2023-03-14coset.png) ![](attachments/2023-03-14-2.png) ![](attachments/2023-03-14aaa.png) ![](attachments/2023-03-14-4.png) See [Shimura variety](Unsorted/Shimura%20variety.md). # Motivations - Why move to defining automorphic forms on double coset spaces instead of on arithmetic quotients? - Reduce complicated problems about automorphic forms to analysis on local groups $\GL_2(\RR)$ and $\GL_2(\QQpadic)$, e.g. studying Hecke operators. - $\GL_2(\QQ)$ is simpler than the collection of arithmetic subgroups of $\SL_2(\ZZ)$ -- e.g. when applying trace formulas, it's easier to understand conjugacy classes of $\GL_2(\QQ)$ than conjugacy class of modular groups $\Gamma(N)$. - The Bruhat decomposition for $\SL_2(\QQ)$ is much simpler than that of $\SL_2(\ZZ)$. # Definitions - An **automorphic form** on an algebraic group $G$ is a function $f\in C^\infty(G, \CC)$ satisfying: - $f$ is right $K_f\dash$finite where $K_f \da \prod_{v< \infty} K_v$ - $f$ is of "uniform moderate growth" - $f$ is $Z(\lieg)\dash$finite. Automorphic functions and relations to [BunG](Unsorted/BunG.md): ![](attachments/Pasted%20image%2020220424193958.png) ![](attachments/Pasted%20image%2020220424194017.png) ![](attachments/Pasted%20image%2020220424194023.png) ![](attachments/2023-03-12-1.png) Where Kloosterman sums appear: ![](attachments/2023-03-12-2.png) # Cusp forms Definition: An automorphic form $f$ on $G$ is called a cusp form if, for any [parabolic](parabolic) $k$ subgroup $P=M N$ of $G$, the $N$-constant term $$ f_{N}(g)=\int_{N(k) \backslash N(\mathbb{A})} f(n g) d n $$ is zero as a function on $G(\mathbb{A})$. Relation to $L^2$ functions and [Schwartz space](Unsorted/Schwartz%20space.md): ![](attachments/Pasted%20image%2020220210183213.png) See also [cuspidal representations](Unsorted/automorphic%20representation#^126feb). # Notes ![](attachments/Pasted%20image%2020220210134403.png) ![](attachments/Pasted%20image%2020220210135137.png) ![](attachments/Pasted%20image%2020220210135159.png) ![](attachments/Pasted%20image%2020220210135649.png) ![](attachments/Pasted%20image%2020220210174710.png) ## Representations From : ![](attachments/Pasted%20image%2020220210174919.png) # Vector-valued For $L$ a lattice and $L'$ its dual, ![](attachments/2023-03-12vec.png) ![](attachments/2023-03-12dasdsa.png) # Fourier expansions