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Periodicities for
Taylor coefficients of half-integral weight modular forms
Pavel Guerzhoy, Michael H. Mertens and Larry Rolen |
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Vol. 307 (2020), No. 1, 137–157
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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 . 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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Authors |
| Pavel Guerzhoy | |
| Department of Mathematics University of Hawaii at Manoa Honolulu, HI United States |
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| Michael H. Mertens | |
| Department of Mathematical
Sciences University of Liverpool Liverpool United Kingdom |
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| Larry Rolen | |
| Department of Mathematics Vanderbilt University Nashville, TN United States |
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