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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Counting zeros in quaternion algebras using Jacobi forms
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by Hati̇ce Boylan, Nils-Peter Skoruppa and Haigang Zhou PDF
Trans. Amer. Math. Soc. 371 (2019), 6487-6509 Request permission

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
  • Hati̇ce 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
  • MR Author ID: 163415
  • Email: nils.skoruppa@gmail.com
  • Haigang Zhou
  • Affiliation: Tongji University, School of Mathematical Sciences, Shanghai, People’s Republic of China
  • MR Author ID: 773510
  • Email: haigangz@tongji.edu.cn
  • 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)
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 6487-6509
  • MSC (2010): Primary 11F50, 11R52
  • DOI: https://doi.org/10.1090/tran/7575
  • MathSciNet review: 3937334