Arithmetic holonomy bounds and their applications

Abstract

On the heels of the proof of the Unbounded Denominators conjecture (previously presented in this seminar by Yunqing Tang), we discuss an upgraded and refined form of our main technical tool in this area, the "arithmetic holonomicity theorem," of which we will detail a proof based on Bost's slopes method. Our treatment will lead us to a new alternative argument for the unbounded denominators theorem on the Fourier expansions of noncongruence modular forms. We will then conclude by explaining how the same arithmetic holonomicity theorem also leads to a proof of the irrationality of all products of two logarithms $\log(1+1/n)\log(1+1/m)$ for arbitrary integer pairs $(n,m)$ with $|1-m/n| < c$, where $c > 0$ is a positive absolute constant. This is a joint work with Frank Calegari and Yunqing Tang.