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

Abstract

The zeta function of an integral lattice $\Lambda$ is the generating function ${\zeta }_{\Lambda }(s)=\sum _{n=0}^{\infty }{a}_n{n}^{-s}$, whose coefficients count the number of left ideals of $\Lambda$ of index $n$. We derive a formula for the zeta function of ${\Lambda }_1\otimes {\Lambda }_2$, where ${\Lambda }_1$ and ${\Lambda }_2$ are $\mathbb{Z}$-orders contained in finite-dimensional semisimple $\mathbb{Q}$-algebras that satisfy a "locally coprime" condition. We apply the formula obtained above to $\mathbb{Z}S\otimes \mathbb{Z}T$ 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 $\mathbb{Z}$-orders $\mathbb{Z}S$ and $\mathbb{Z}T$ are locally coprime and their zeta functions are known.

arXiv comment14 pages

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