back to N-Cube Days XXII

Wushi Goldring (Stockholm University)

Title: Understanding the algebraic nature of automorphic representations via functoriality and group theory

Abstract: My talk is motivated by two questions – One specific, one broader:
(1) Specific: Which automorphic representations admit algebraic Hecke eigenvalues?
(2) Broader: What mathematics can be understood, reinterpreted or even generated by groups?
In particular, we view (1) as a key test-case of (2).

Automorphic representations are infinite-dimensional representations of reductive groups over number fields, defined using harmonic analysis. A concrete example of their real-analytic nature is Maass forms: non-holomorphic analogues of modular forms on the complex upper half-plane. For every prime p, they admit p-adic analogues of Laplacian eigenvalues called Hecke eigenvalues. A major mystery of the Langlands Program is that some automorphic representations have algebraic Hecke eigenvalues; others have transcendental ones. For some, the algebraicity follows from algebraic geometry/topology, while for others there are conjectures predicting either algebraicity or transcendence. But there is also a ‘grey zone’ where it is unclear what to expect.

I will explain progress on the mystery by using Langlands functoriality and group theory to study two fundamental invariants of automorphic representations – the infinitesimal character and - roughly - complex conjugation. We introduce two dichotomies on infinitesimal characters that go beyond the L, C and W-algebraic of Buzzard-Gee and Patrikis: D vs M and D^Gal vs M^Gal-algebraic. Using the complex conjugation, we give group-theoretic obstructions and positive examples regarding when the algebraicity of Hecke eigenvalues is reducible via Langlands functoriality to a case known by geometry. In particular, the obstructions give a conceptual explanation for why no such reduction exists for L-algebraic Maass forms. The positive examples exhibit new cases of algebraic Hecke eigenvalues in settings – like Maass forms – where no direct link to geometry is known. For some of them, we also construct the Galois representations predicted by the Langlands correspondence.