The genus zeta function of hereditary orders in central simple algebras over global fields

Abstract

Louis Solomon introduced the notion of a zeta function ζ Θ ( s ) {\zeta _\Theta }(s) of an order Θ \Theta in a finite-dimensional central simple K-algebra A, with K a number field or its completion K P {K_P} (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate ζ Θ ( s ) {\zeta _\Theta }(s) , it is sufficient to consider the zeta function of local orders Θ P {\Theta _P} . However, since these local orders Θ P {\Theta _P} are in general not principal ideal domains, their zeta function is a finite sum of so-called ’partial zeta functions’. The most complicated term is the ’genus zeta function’, Z Θ P ( s ) {Z_{{\Theta _P}}}(s) , which is related to the free Θ P {\Theta _P} -ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., [ A : K ] = 4 [A:K] = 4 ). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function Z Θ P ( s ) {Z_{{\Theta _P}}}(s) of any local hereditary order Θ P {\Theta _P} in a central simple algebra over a local field. We obtain Z Θ P ( s ) {Z_{{\Theta _P}}}(s) as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.