Zeta functions and ideal classes of quaternion orders (Q2409599)

The goal of this paper is to show that, given a totally definite quaternion algebra \(A\) over a number field \(F\) with a maximal \(\mathcal{O}_F\)-order \(\Lambda\), the number of left ideal classes in \(\Lambda\) is finite by means of the zeta function \(\zeta_\Lambda(s)\), an analogue of the Dedekind zeta function of a number field. The strategy is similar to Stark's proof of the finiteness of the number of ideal classes in a number field in [\textit{H.M. Stark}, Bull. Am. Math. Soc. 81, 961--972 (1975; Zbl 0329.12010)]. In fact, \(\zeta_\Lambda\) can be written as the sum of partial zeta functions associated to the left ideal classes in \(\Lambda\). Such partial zeta functions can be expressed in terms of Epstein zeta functions of quadratic forms related to the reduced norm on \(A\). Since the value of the Epstein zeta function at \(s=2\) can be estimated in terms of the Gamma function and the discriminant of the associated quadratic form, which is \(1\) in our case, the values of the partial zeta functions at \(s=2\) are all greater than a positive constant independent of the ideal classes. Since they sum up to the zeta function which converges at \(s=2\), \(\Lambda\) can have only finitely many left ideal classes. The technique involved in this proof also serves to give an alternative proof of Eichler's mass formula by considering the behavior of the zeta function and the partial zeta functions at \(s=0\). This also leads to an alternative proof of the fact that the number of ramified primes in \(A\) is even.