Solomon's second conjecture: A proof for local hereditary orders in central simple algebras

Abstract

Solomon’s second conjecture concerns partial zeta functions of local orders. The exact definitions of orders and ideals can be found in [R], and we use the partial zera functions, as defined in [BR]. We briefly recall the definitions. Let R, be a discrete valuation ring in a local field K with valuation v. A local order 0 is an R,-order in a central simple K-algebra A = M,(D). If IZ = 1 the valuation u has a unique extension vD in the skew field D. We denote with d the unique maximal R,-order in D and with fi the maximal ideal in A. Then # (A/b) = q is finite. We fix a uniformizing element n of A, i.e., uD(rc) = 1. If n # 1 the local orders are not so easy to describe. The maximal R,-orders /i in A are isomorphic with M,(A). The partial zeta functions of a local order 0 correspond to a choice of two left @-ideals 2, -4 in A (cf. [BR, p. 1371):