The Unbounded Denominators Conjecture - Yunqing Tang

Explore the unbounded denominators conjecture in this joint IAS/Princeton University Number Theory Seminar talk. Delve into the proof of this conjecture, first proposed by Atkin and Swinnerton-Dyer, which states that a modular form for a finite index subgroup of SL2(ℤ) with Fourier coefficients having bounded denominators must be a modular form for a congruence subgroup. Follow speaker Yunqing Tang from Princeton University as she outlines the proof based on a new arithmetic algebraization theorem, developed in collaboration with Frank Calegari and Vesselin Dimitrov. Gain insights into module forms, bounded denominators, limitations, boundary conditions, and the intricacies of the proof through topics such as gender modules and disc cubes.

Introduction
Module form
Bounded denominator
Module forms
Limitations
Boundary
Gender module
Disc cube
Proof