Periodicities for Taylor coefficients of half-integral weight modular forms

Pavel Guerzhoy, Michael H. Mertens and Larry Rolen

Vol. 307 (2020), No. 1, 137–157

DOI: 10.2140/pjm.2020.307.137

Abstract

Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper half-plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around ${\tau }_0=i$ of the classical Jacobi theta function ${𝜃}_3$. Here, we generalize the phenomenon observed by Romik to a broader class of modular forms of half-integral weight and, in particular, prove the conjecture.

Keywords

modular forms, Taylor coefficients, $q$-expansion principle, congruences

Mathematical Subject Classification 2010

Primary: 11F25, 11F33, 11F37

Milestones

Received: 10 September 2019

Accepted: 5 March 2020

Published: 8 August 2020