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Periodicities for Taylor coefficients of half-integral weight modular forms

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Mertens,  Michael H.
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Guerzhoy, P., Mertens, M. H., & Rolen, L. (2020). Periodicities for Taylor coefficients of half-integral weight modular forms. Pacific Journal of Mathematics, 307(1), 137-157. doi:10.2140/pjm.2020.307.137.


Cite as: https://hdl.handle.net/21.11116/0000-0009-DC94-8
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 $\tau=i$ of the classical Jacobi theta
function. Here, we prove this conjecture and generalize the phenomenon observed
by Romik to a general class of modular forms of half-integral weight.