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Abstract
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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
of the classical Jacobi
theta function
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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.
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Keywords
modular forms, Taylor coefficients, $q$-expansion
principle, congruences
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Mathematical Subject Classification 2010
Primary: 11F25, 11F33, 11F37
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Milestones
Received: 10 September 2019
Accepted: 5 March 2020
Published: 8 August 2020
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