Counting zeros in quaternion algebras using Jacobi forms

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 ( 4 n − r 2 ) 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 2 on Γ 0 ( N ) \Gamma _0(N) and for the action of Hecke operators on Jacobi theta series associated to maximal orders of definite quaternion algebras.