Trace formulas for the norm one group of totally definite quaternion algebras

TL;DR

This paper extends classical trace formulas and class number formulas from Eichler and Pizer to the norm one group of totally definite quaternion algebras, incorporating new invariants and spinor selectivity theory.

Contribution

It introduces a class number formula for the norm one group of quaternion algebras with residually unramified orders, and refines Eichler's class number formula by including order-specific information.

Findings

Derived a class number formula for the norm one group of quaternion algebras.

Provided a refined formula for the number of ideal classes in the spinor class of orders.

Utilized auxiliary invariants and spinor selectivity theory in the formulas.

Abstract

In his pioneering work [Crelle's Journal, 1955], Eichler established the theory of trace formulas for Brandt matrices of quaternion orders. From it he derived a class number formula for Eichler orders in a totally definite quaternion algebra $D$. Extending Eichler's work, Pizer [Crelle's Journal, 1973] proved a formula for the type number of Eichler orders in $D$. In this paper, we extend their results to the norm one group of $D$. More precisely, we present a class number formula for the norm one group of $D$ with respect to a class of orders $\mathcal{O}$, called \emph{residually unramified orders}, which includes all Eichler orders. Our second result gives a formula for the number of ideal classes in the spinor class of $\mathcal{O}$, which refines Eichler's class number formula. It is worth mentioning that these class number formulas not only depend on the genus of orders as Eichler and Pizer's formulas, but also depend on the orders themselves. We introduce certain auxiliary invariants in order to keep track of the global information on the relationship between certain CM orders and $\mathcal{O}$, and use them to describe our formulas. Both our class number formulas make use of the optimal spinor selectivity theory for quaternion orders.