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

TL;DR

This paper derives a formula for the zeta function of tensor products of certain integral algebras, enabling computation of zeta functions for adjacency algebras of product association schemes under specific coprimality conditions.

Contribution

It introduces a new formula for the zeta function of tensor products of locally coprime integral algebras and applies it to adjacency algebras of association schemes.

Findings

Derived a formula for zeta functions of tensor products of integral algebras.

Applied the formula to adjacency algebras of product association schemes.

Computed explicit zeta functions in several coprimality cases.

Abstract

The zeta function of an integral lattice $\Lambda$ is the generating function $\zeta_{\Lambda}(s) = \sum\limits_{n=0}^{\infty} a_n n^{-s}$, 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.