Periodicities for Taylor coefficients of half-integral weight modular forms

Pavel Guerzhoy, Michael H. Mertens, Larry Rolen

Type: Article

Publication Date: 2020-08-08

Citations: 2

DOI: https://doi.org/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 $\tau=i$ 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.

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Pacific Journal of Mathematics - View
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+	Berechnung von Zetafunktionen an ganzzahligen Stellen	1979	Carl Ludwig Siegel K. Chandrasekharan H. Maaß
+	Elliptic Modular Forms and Their Applications	2008	Don Zagier
+ PDF Chat	A Generalized Jacobi Theta Function and Quasimodular Forms	1995	Masanobu Kaneko Don Zagier
+	Modular Forms of Half Integral Weight	2009	Goro Shimura
+ PDF Chat	Ueber die Entwickelungscoefficienten der lemniscatischen Functionen	1898	Adolf Hurwitz
+	Modular forms associated to real quadratic fields	1975	Don Zagier
+	The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers	1975	Nicholas M. Katz
+	Sums involving the values at negative integers of L-functions of quadratic characters	1975	Henri Cohen
+	Einheitliche Begründung der Vollständigkeitssätze für die Poincaréschen Reihen von reeller Dimension bei beliebigen Grenzkreisgruppen von erster Art	1941	Hans Petersson
+	p-adic interpolation of Taylor coefficients of modular forms	2007	Boris A. Datskovsky Pavel Guerzhoy