Solomon's conjectures and the local functional equation for zeta functions of orders

p and introduced a "global" zeta function fA(s), which depends on A and R but not on A, with the property that the P-part of ÇA(s) coincides with ?AJ>() f° r almost all P. (To be explicit, this occurs whenever Ap is a direct sum of full matrix algebras over fields, and Ap is a maximal .Rp-order in Ap.) Solomon's conjectures involve the comparison between ÇA(s) and ÇA(s) at arbitrary P's. Let us place the above in the more general setting used by Solomon. Let L be a full A-lattice in an A -module V, and define fc(')= Z (L:M)~\ MCL where M ranges over all full A-sublattices of L. To define the "global" function Jy(s), we start with the Wedderburn decomposition of A: