Zeta functions for tensor products of locally coprime integral adjacency algebras of association schemes

ABSTRACTThe zeta function of an integral lattice Λ is the generating function ζΛ(s)=∑n=0∞ann−s, whose coefficients count the number of left ideals of Λ of index n. We derive a formula for the zeta function of Λ1⊗Λ2, where Λ1 and Λ2 are ℤ-orders contained in finite-dimensional semisimple ℚ-algebras that satisfy a “locally coprime” condition. We apply the formula obtained above to ℤS⊗ℤT and obtain the zeta function of the adjacency algebra of the direct product of two finite association schemes (X,S) and (Y,T) in several cases where the ℤ-orders ℤS and ℤT are locally coprime and their zeta functions are known.