Zeta functions of adjacency algebras of association schemes of prime order or rank two

Abstract

For a module $L$ which has only finitely many submodules with a given finite index we define the zeta function of $L$ to be a formal Dirichlet series $\zeta_L(s)=\sum_{n\geq 1}a_nn^{-s}$ where $a_n$ is the number of submodules of $L$ with index $n$. For a commutative ring $R$ and an association scheme $(X,S)$ we denote the adjacency algebra of $(X,S)$ over $R$ by $RS$. In this article we aim to compute $\zeta_{\mathbb{Z}S}(s)$, where $\mathbb{Z}S$ is viewed as a regular $\mathbb{Z}S$-module, under the assumption that $|X|$ is a prime or $|S|=2$.