Vol. 307, No. 1, 2020

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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
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 τ0 = i of the classical Jacobi theta function 𝜃3. 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
Authors
Pavel Guerzhoy
Department of Mathematics
University of Hawaii at Manoa
Honolulu, HI
United States
Michael H. Mertens
Department of Mathematical Sciences
University of Liverpool
Liverpool
United Kingdom
Larry Rolen
Department of Mathematics
Vanderbilt University
Nashville, TN
United States