# AWS 2022: Automorphic Forms Beyond $\GL_2$

Tags:
#projects/active 

Refs:
[Langlands](Unsorted/Langlands.md)
[Zhiwei Yun project group](Zhiwei%20Yun%20project%20group.md)

# Resources
- AWS website: <https://www.math.arizona.edu/~swc/>
- My notes project: [Automorphic Forms and Langlands Kevin Buzzard Notes](Projects/2022%20AWS/Automorphic%20Forms%20and%20Langlands%20Kevin%20Buzzard%20Notes.md)
- Geometric Langlands talks: <https://math.uchicago.edu/~amathew/GL2021-22.html>
- To read: see <https://www.math.arizona.edu/~swc/aws/2022/2022EischenNotes.pdf>
![](attachments/Pasted%20image%2020220210131726.png)

# Motivating problems

- Algebraicity or rationality of special values of [automorphic](automorphic) $L\dash$functions
	- E.g. Euler's proof that $\zeta(2 k)=(-1)^{k+1} \pi^{2 k} \frac{B_{2 k}}{2(2 k) !}$ is algebraic and in fact rational up to a transcendental factor.
- [Bloch-Kato](Unsorted/Bloch-Kato.md) conjecture
- [Gan–Gross–Prasad conjecture](Gan–Gross–Prasad%20conjecture)
-  [Iwasawa main conjecture](Iwasawa%20main%20conjecture) for $\GL_2$
- Understanding automorphic forms on unitary groups (to motivate $p\dash$adic analogs later)
- Shimura's proof in [Shi75] of algebraicity of certain values of the Rankin-Selberg convolution, i.e. the *Rankin–Selberg method*.
- [Ramanujan-Petersson conjecture](Ramanujan-Petersson%20conjecture.md)
- [Arthur's conjecture](Arthur's%20conjecture.md)
- [Adam's conjecture](Adam's%20conjecture)
- [Selberg 1/4 conjecture](Selberg%201/4%20conjecture) (open)

# Topics

- [Unsorted/admissible representation](Unsorted/admissible%20representation.md)
- [Hecke character](Hecke%20character)
- [p-adic zeta function](p-adic%20zeta%20function.md)
- [cusp form](cusp%20form.md)
- [automorphic form](Unsorted/automorphic%20form.md)
- The [Hecke algebra](Hecke%20algebra)
- The [Tate curve](Tate%20curve).
- [Siegel modular forms](Siegel%20modular%20forms)
- The [Hodge bundle](Hodge%20bundle)
- Maass–Shimura operators
- Rankin–Cohen brackets
- [Hecke eigenform](Hecke%20eigenform)
- [semisimple](semisimple) [reductive](Unsorted/reductive.md) linear [algebraic groups](Unsorted/algebraic%20group.md)
- [adelic group](adelic%20group.md)
- The [Satake isomorphism](Satake%20isomorphism)
- [Principal series representations](Principal%20series%20representations.md)
- [Hermitian spaces](Unsorted/Hermitian.md)
- [local theta correspondence](local%20theta%20correspondence.md)
- [Tate's thesis](Tate's%20thesis)
- [Steinberg representation](Steinberg%20representation)
- [Shimura variety](Unsorted/Shimura%20variety.md)
- New terms:
	- [tempered](tempered) representations
	- [packets](packets)
	- [theta lifts](theta%20lifts)
	- [cusp forms](cusp%20forms)
	- [quasi-split](quasi-split)
	- [contragredient](contragredient)

## Background

- Arithmetic geometry
	- The [Hasse principle](Unsorted/Hasse%20principle.md)
- ANT:
	- [Ideles](Unsorted/Ideles.md)
	- [field norm](Unsorted/field%20norm.md)
	- [split](Unsorted/ramification%20index.md), [inert](Unsorted/ramification%20index.md), [ramified](Unsorted/ramification%20index.md)
	- [place](Unsorted/Valuations.md)
- Modular forms:
	- [Artin L function](Artin%20L%20function)
	- [modular form](Unsorted/modular%20form.md)
- Group cohomology
	- [invariants](Unsorted/group%20cohomology.md) and [coinvariants](Unsorted/group%20cohomology.md)
	- [Frobenius reciprocity](Frobenius%20reciprocity)
- Algebraic groups
	- [root system](root%20system)
	- [Levi decomposition](Levi%20decomposition)
	- [parabolic](Unsorted/parabolic.md) and [Borel](Borel)
	- [unipotent radical](unipotent%20radical)
	- 
- Misc
	- [Langlands dual](Langlands%20dual)

# Notes

- [Fermat's Last Theorem](Fermat's%20Last%20Theorem):
	A prime $p$ does not divide the class number of the cyclotomic field $\mathbb{Q}\left(e^{2 \pi i / p}\right)$ if and only if $p$ does not divide the numerators of the Bernoulli numbers $B_{2}, B_{4}, \ldots, B_{p-3}$.  This leads to proofs of special cases of Fermat's Last Theorem.
- Rationality of [L functions](Unsorted/L%20function.md): try to reduce to a question about [Eisenstein series](Unsorted/Eisenstein%20series.md).
	Strategy:
![](attachments/Pasted%20image%2020220210180347.png)
- Twisted coinvariant spaces:
![](attachments/Pasted%20image%2020220210225134.png)