- Tags
-
Refs:
- https://www.math.arizona.edu/~swc/aws/2022/2022EischenNotes.pdf
- Introduction to the plectic conjecture https://webusers.imj-prg.fr/~jan.nekovar/pu/introplec.pdf#page=1 #resources/notes
- Notes on Fourier analysis on number fields: https://people.math.harvard.edu/~sli/math99r_f20/index.html #resources/course-notes
-
Links:
- modular form
- automorphic representation
- Shimura variety
- Hecke stack
- Shimura--Taniyama--Weil conjecture
- Fermat’s Last Theorem
automorphic form
-
Rewriting the half-plane as a homogeneous space: let \({\operatorname{SL}}_2({\mathbf{R}})\curvearrowright{\mathbb{H}}\) on the right by \(z\curvearrowleft{ \begin{bmatrix} {a} & {b} \\ {c} & {d} \end{bmatrix} }\coloneqq{ax + b\over cx+d}\), then the stabilizer at \(i\) is \({\operatorname{Stab}}_{{\operatorname{SL}}_2({\mathbf{R}})}(i) = \left\{{{ \begin{bmatrix} {a} & {b} \\ {-b} & {a} \end{bmatrix} }}\right\} = {\operatorname{SO}}_2({\mathbf{R}})\). So \begin{align*} {\mathbb{H}}\cong \dcoset{1}{{\operatorname{SL}}_2({\mathbf{R}})}{{\operatorname{Stab}}_{{\operatorname{SL}}_2({\mathbf{R}})}(i) } = \dcoset{1}{{\operatorname{SL}}_2({\mathbf{R}})}{{\operatorname{SO}}_2({\mathbf{R}}) } \end{align*}
- Note that \({\operatorname{SO}}_2({\mathbf{R}}) \hookrightarrow{\operatorname{SL}}_2({\mathbf{R}})\) is the maximal compact subgroup.
-
Write \({\mathbb{H}}= \dcoset{{\operatorname{SL}}_2({\mathbf{Z}})}{{\operatorname{SL}}_2({\mathbf{R}})}{1}\) or \(\dcoset{1}{{\operatorname{SL}}_2({\mathbf{R}})}{{\operatorname{SO}}_2({\mathbf{Q}})}\), so \begin{align*} X = \dcoset{{\operatorname{SL}}_2({\mathbf{Z}})}{{\operatorname{SL}}_2({\mathbf{R}})}{{\operatorname{SO}}_2({\mathbf{Q}})} \end{align*}
-
More generally, replace \(\dcoset{\Gamma}{{\mathbb{H}}}{1}\) with \(\dcoset{{\mathbf{Q}}}{{\mathbf{A}}}{1}\), i.e. \begin{align*} X = \dcoset{\operatorname{GL}_n({\mathbf{Q}})}{\operatorname{GL}_n({\mathbf{A}})}{1} \end{align*}
-
Not sure how this relates yet: a moduli space of unimodular lattices is \begin{align*} X = \dcoset{{\operatorname{SL}}_2(\widehat{{\mathbf{Z}}})}{{\operatorname{SL}}_2({\mathbf{A}}_{\mathbf{Q}})}{{\operatorname{SL}}_2({\mathbf{Q}})} \cong \dcoset{1}{{\operatorname{SL}}_2({\mathbf{R}})}{{\operatorname{SL}}_2({\mathbf{Z}})} \end{align*}
-
Write the space of automorphic functions as \begin{align*} {\mathcal{A}}\coloneqq C^\infty\qty{\dcoset{\operatorname{GL}_n(\widehat{{\mathbf{Z}}})}{\operatorname{GL}_n({\mathbf{A}}_{{\mathbf{Q}}})}{\operatorname{GL}_n({\mathbf{Q}}) )} \longrightarrow{\mathbf{C}}} \end{align*}
-
Let \(K = { \mathbf{F} }_q[C]\) be the function field associated to \(C\) a curve, so \(\operatorname{ff}(K) = { \mathbf{F} }_q(C)\).
-
Another common setup: can realize certain affine modular curves \(\dcoset{\Gamma}{{\mathbb{H}}}{1}\) as \(\dcoset{\operatorname{GL}_2({\mathbf{Q}})}{\operatorname{GL}_2({\mathbf{A}}_{\mathbf{Q}})}{K}\) where \(K = K^\infty \times K_\infty\) and \(K^\infty \leq \operatorname{GL}_2({\mathbf{A}}_{\mathbf{Q}}^\infty)\) is a compact open and \(K_\infty \coloneqq{\mathbf{R}}_{>0}\times {\operatorname{SO}}_2({\mathbf{R}})\).
-
\(\operatorname{GL}_2({\mathbf{A}}_{\mathbf{Q}}) \leq \operatorname{GL}_2({\mathbf{R}}) \times \prod'_{p\in {\left\lvert {\operatorname{Spec}{\mathbf{Z}}} \right\rvert}} \operatorname{GL}_2({ {\mathbf{Q}}_p })\) where almost every entry is in \(\operatorname{GL}_2({ {\mathbf{Z}}_{\widehat{p}} })\). An automorphic form is a function on \(\dcoset{\operatorname{GL}_2({\mathbf{Q}})}{\operatorname{GL}_2({\mathbf{A}})}{1}\), instead of things like \(\dcoset{\Gamma}{X}{1}\) where \(X = {\mathbb{H}}, {\operatorname{SL}}_2({\mathbf{R}}), \operatorname{GL}_2({\mathbf{R}})\), etc.
-
\({\mathbf{Q}}\hookrightarrow{\mathbf{A}}_{\mathbf{Q}}\) is discrete, as is \(\operatorname{GL}_2({\mathbf{Q}})\hookrightarrow\operatorname{GL}_2({\mathbf{A}})\).
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 \(\operatorname{GL}_2({\mathbf{R}})\) and \(\operatorname{GL}_2({ {\mathbf{Q}}_p })\), e.g. studying Hecke operators.
- \(\operatorname{GL}_2({\mathbf{Q}})\) is simpler than the collection of arithmetic subgroups of \({\operatorname{SL}}_2({\mathbf{Z}})\) – e.g. when applying trace formulas, it’s easier to understand conjugacy classes of \(\operatorname{GL}_2({\mathbf{Q}})\) than conjugacy class of modular groups \(\Gamma(N)\).
- The Bruhat decomposition for \({\operatorname{SL}}_2({\mathbf{Q}})\) is much simpler than that of \({\operatorname{SL}}_2({\mathbf{Z}})\).
Definitions
-
An automorphic form on an algebraic group \(G\) is a function \(f\in C^\infty(G, {\mathbf{C}})\) satisfying:
- \(f\) is right \(K_f{\hbox{-}}\)finite where \(K_f \coloneqq\prod_{v< \infty} K_v\)
- \(f\) is of “uniform moderate growth”
- \(f\) is \(Z({\mathfrak{g}}){\hbox{-}}\)finite.
Automorphic functions and relations to BunG: 
Cusp forms
Definition: An automorphic form \(f\) on \(G\) is called a cusp form if, for any parabolic \(k\) subgroup \(P=M N\) of \(G\), the \(N\)-constant term \begin{align*} f_{N}(g)=\int_{N(k) \backslash N(\mathbb{A})} f(n g) d n \end{align*} is zero as a function on \(G(\mathbb{A})\).
Relation to \(L^2\) functions and Schwartz space: 
See also cuspidal representations.
Notes
Representations
From https://www.math.arizona.edu/~swc/aws/2022/2022EischenNotes.pdf: 
