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Automorphic representations and Harmonic analysis
Automorphic representations and Harmonic analysis
Counting cuspidal automorphic representations over a function field
Counting cuspidal automorphic representations over a function field
Organizers
Speaker
Hongjie Yu
Time
Thursday, October 15, 2026 1:30 PM - 2:30 PM
Venue
A3-4-301
Online
Zoom 435 529 7909
(BIMSA)
Abstract
In 1981, Drinfeld counted the irreducible rank-two ℓ-adic local systems on a smooth projective curve over a finite field that are fixed by the Frobenius endomorphism. Strikingly, the resulting number has the shape of a point count on a variety over a finite field, pointing to a hidden geometric structure. Deligne later proposed conjectures aimed at extending and conceptualizing Drinfeld’s enumeration. Through the Langlands correspondence for function fields, this enumeration is equivalent to counting certain cuspidal automorphic representations. Ngô Bảo Châu observed that trace formulas over a Lie algebra compute point counts on the moduli space of Higgs bundles over a curve over a finite field. Using this observation, we connect such enumerative problems to counts of stable Higgs bundles via Arthur’s non-invariant trace formula, thereby confirming some of Deligne’s conjectures.