Tags: #projects/active
Refs: Langlands Zhiwei Yun project group
Resources
 AWS website: https://www.math.arizona.edu/~swc/
 My notes project: Automorphic Forms and Langlands Kevin Buzzard Notes
 Geometric Langlands talks: https://math.uchicago.edu/~amathew/GL202122.html
 To read: see https://www.math.arizona.edu/~swc/aws/2022/2022EischenNotes.pdf
Motivating problems

Algebraicity or rationality of special values of automorphic \(L{\hbox{}}\)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.
 BlochKato conjecture
 Gan–Gross–Prasad conjecture
 Iwasawa main conjecture for \(\operatorname{GL}_2\)
 Understanding automorphic forms on unitary groups (to motivate \(p{\hbox{}}\)adic analogs later)
 Shimura’s proof in [Shi75] of algebraicity of certain values of the RankinSelberg convolution, i.e. the Rankin–Selberg method.
 RamanujanPetersson conjecture
 Arthur’s conjecture
 Adam’s conjecture
 Selberg 1/4 conjecture (open)
Topics
 Unsorted/admissible representation
 Hecke character
 padic zeta function
 cusp form
 automorphic form
 The Hecke algebra
 The Tate curve.
 Siegel modular forms
 The Hodge bundle
 Maass–Shimura operators
 Rankin–Cohen brackets
 Hecke eigenform
 semisimple reductive linear algebraic groups
 adelic group
 The Satake isomorphism
 Principal series representations
 Hermitian spaces
 local theta correspondence
 Tate’s thesis
 Steinberg representation
 Shimura variety

New terms:
 tempered representations
 packets
 theta lifts
 cusp forms
 quasisplit
 contragredient
Background

Arithmetic geometry
 The Hasse principle
 ANT:

Modular forms:
 Artin L function
 modular form

Group cohomology
 invariants and coinvariants
 Frobenius reciprocity

Algebraic groups
 root system
 Levi decomposition
 parabolic and Borel
 unipotent radical

Misc
 Langlands dual
Notes
 Fermat’s Last Theorem: 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 \(B_{2}, B_{4}, \ldots, B_{p3}\). This leads to proofs of special cases of Fermat’s Last Theorem.
 Rationality of L functions: try to reduce to a question about Eisenstein series. Strategy:
 Twisted coinvariant spaces: