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Traces of Eichler-Brandt matrices and type numbers of quaternion orders. (English) Zbl 0651.12011

Let A be a totally definite quaternion algebra over a totally real algebraic number field and M be the ring of algebraic integers of F. For any M-order G of A formulas are derived for the mass m(G) and the type number t(G) and for the trace of the Eichler-Brandt matrix B(G,J) (where J is an integral ideal) in terms of genus invariant of G and of invariants of F and J. Thus earlier results of Eichler and others are generalized. The proofs use integration on some quotient spaces of the idele group of A and an elementary version of Selberg’s trace formula.
Reviewer: M.Peters

MSC:

11R52 Quaternion and other division algebras: arithmetic, zeta functions
11E41 Class numbers of quadratic and Hermitian forms
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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