Counting zeros in quaternion algebras using Jacobi forms

Boylan, Hatice; Skoruppa, Nils-Peter; Zhou, Haigang

doi:10.1090/tran/7575

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Counting zeros in quaternion algebras using Jacobi forms

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Boylan, Hatice
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1090/tran/7575
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http://www.mpim-bonn.mpg.de/preblob/5665
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Citation

Boylan, H., Skoruppa, N.-P., & Zhou, H. (2019). Counting zeros in quaternion algebras using Jacobi forms. Transactions of the American Mathematical Society, 371(9), 6487-6509. doi:10.1090/tran/7575.

Cite as: https://hdl.handle.net/21.11116/0000-0004-D5DD-3

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 Γ_0(N) and for the action of Hecke operators on Jacobi theta series associated to maximal orders of definite
quaternion algebras.