Petersson inner products of weight-one modular forms

Viazovska, Maryna

doi:10.1515/crelle-2016-0042

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Petersson inner products of weight-one modular forms

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Viazovska, Maryna
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1515/crelle-2016-0042
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https://doi.org/10.48550/arXiv.1211.4715
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Citation

Viazovska, M. (2019). Petersson inner products of weight-one modular forms. Journal für die Reine und Angewandte Mathematik, 749, 133-159. doi:10.1515/crelle-2016-0042.

Cite as: https://hdl.handle.net/21.11116/0000-0004-F859-1

Abstract

n this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product
is equal to the logarithm of a certain algebraic number lying in a ring class field associated
to the binary quadratic form. A similar result was obtained independently using a different
method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization
formula for the algebraic number obtained by exponentiating a Petersson product.