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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Counting zeros in quaternion algebras using Jacobi forms


Authors: Hatice Boylan, Nils-Peter Skoruppa and Haigang Zhou
Journal: Trans. Amer. Math. Soc. 371 (2019), 6487-6509
MSC (2010): Primary 11F50, 11R52
DOI: https://doi.org/10.1090/tran/7575
Published electronically: October 17, 2018
MathSciNet review: 3937334
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Abstract: We use the theory of Jacobi forms to study the number of elements in a maximal order of a definite quaternion algebra over the field of rational numbers whose characteristic polynomial equals a given polynomial. A certain weighted average of such numbers equals (up to some trivial factors) the Hurwitz class number $ H(4n-r^2)$. As a consequence we obtain new proofs for Eichler's trace formula and for formulas for the class and type number of definite quaternion algebras. As a secondary result we derive explicit formulas for Jacobi Eisenstein series of weight $ 2$ on $ \Gamma _0(N)$ and for the action of Hecke operators on Jacobi theta series associated to maximal orders of definite quaternion algebras.


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Additional Information

Hatice Boylan
Affiliation: İstanbul Üniversitesi, Fen Fakültesi, Matematik Bölümü, 34134 Vezneciler, İstanbul, Turkey
Email: hatice.boylan@gmail.com

Nils-Peter Skoruppa
Affiliation: Universität Siegen, Siegen, Germany; and Tongji University, School of Mathematical Sciences, Shanghai, People’s Republic of China
Email: nils.skoruppa@gmail.com

Haigang Zhou
Affiliation: Tongji University, School of Mathematical Sciences, Shanghai, People’s Republic of China
Email: haigangz@tongji.edu.cn

DOI: https://doi.org/10.1090/tran/7575
Keywords: Jacobi forms, arithmetic of quaternion algebras
Received by editor(s): May 23, 2017
Received by editor(s) in revised form: December 5, 2017
Published electronically: October 17, 2018
Additional Notes: The first author thanks the Max-Planck Institut für Mathematik in Bonn for inviting her for a research stay in 2016/2017, part of which was used for the preparation of this article.
The third author is supported by the National Natural Science Foundation of China (Grant No. 11271283)
Article copyright: © Copyright 2018 American Mathematical Society