Tags: #projects/active

Refs: Langlands

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/GL2021-22.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.
• Bloch-Kato 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 Rankin-Selberg convolution, i.e. the Rankin–Selberg method.
• Ramanujan-Petersson conjecture
• Arthur's conjecture
• Adam's conjecture

Topics

• Admissible representation
• Hecke character
• p-adic 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
• New terms:
  • tempered representations
  • packets
  • theta lifts
  • cusp forms
  • quasi-split

Background

• Arithmetic geometry
  • The Hasse Principle
• ANT:
  • Ideles
  • field norm
  • split, inert, ramified
  • place
• 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 \(B_{2}, B_{4}, \ldots, B_{p-3}\). 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: