AWS 2022: Automorphic Forms Beyond \(\operatorname{GL}_2\)

Tags: #projects/active

Refs: Langlands Zhiwei Yun project group


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
  • Selberg 1/4 conjecture (open)


  • Unsorted/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
  • Steinberg representation
  • Shimura variety
  • New terms:
    • tempered representations
    • packets
    • theta lifts
    • cusp forms
    • quasi-split
    • contragredient



  • 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: attachments/Pasted%20image%2020220210180347.png
  • Twisted coinvariant spaces: attachments/Pasted%20image%2020220210225134.png
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