The Unbounded Denominators Conjecture

Abstract: The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer in 1968, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. Our proof of this conjecture is based on a new arithmetic algebraization theorem, which has its root in the classical Borel—Dwork rationality criterion. In this talk, we will discuss some ingredients in the proof and a variant of our arithmetic algebraization theorem, which we will use to prove the irrationality of certain 2-adic zeta value.

This is joint work with Frank Calegari and Vesselin Dimitrov.