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Signs of Fourier coefficients of half-integral weight modular forms

Lester, Stephen and Radziwiłł, Maksym (2021) Signs of Fourier coefficients of half-integral weight modular forms. Mathematische Annalen, 379 (3-4). pp. 1553-1604. ISSN 0025-5831. doi:10.1007/s00208-020-02123-0.

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Let g be a Hecke cusp form of half-integral weight, level 4 and belonging to Kohnen’s plus subspace. Let c(n) denote the nth Fourier coefficient of g, normalized so that c(n) is real for all n≥1. A theorem of Waldspurger determines the magnitude of c(n) at fundamental discriminants n by establishing that the square of c(n) is proportional to the central value of a certain L-function. The signs of the sequence c(n) however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that c(n)<0 and respectively c(n)>0 holds for a positive proportion of fundamental discriminants n. Moreover we show that the sequence {c(n)} where n ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of c(n) which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms g of level 4N with N odd, square-free.

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Lester, Stephen0000-0003-1977-9205
Additional Information:© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Received 26 April 2019; Revised 21 August 2020; Accepted 19 November 2020; Published 09 January 2021. We would like to thank Abhishek Saha for many helpful conversations regarding Kohnen’s plus space and for pointing out Shimura’s paper [28]. We would also like to express our gratitude to Dinakar Ramakrishnan who provided us with comments on an earlier draft of the appendix. Finally, we would like to thank the anonymous referee for a careful reading of the paper.
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King’s College LondonUNSPECIFIED
Issue or Number:3-4
Record Number:CaltechAUTHORS:20210122-132850144
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Official Citation:Lester, S., Radziwiłł, M. Signs of Fourier coefficients of half-integral weight modular forms. Math. Ann. 379, 1553–1604 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:107662
Deposited By: Tony Diaz
Deposited On:22 Jan 2021 21:40
Last Modified:01 Apr 2021 18:00

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