The number of ideals of $\mathbb{Z}[x]$ containing $x(x-\alpha )(x-\beta )$ with given index

Abstract

It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $⟨B⟩$ denote the subring of the full matrix ring generated by $B$. Then $⟨B⟩$ is a free $\mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $⟨B⟩$ with given finite index. Thus, the formal Dirichlet series ${\zeta }_{⟨B⟩}(s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$ is well-defined where $a_n$ is the number of ideals of $⟨B⟩$ with index $n$. In this article we aim to find an explicit form of ${\zeta }_{⟨B⟩}(s)$ when $B$ has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, $⟨B⟩$ is isomorphic to $\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$ where $m(x)$ is the minimal polynomial of $B$ over $\mathbb{Q}$, and $\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$ is isomorphic to $\mathbb{Z}[x]/m(x+\gamma )\mathbb{Z}[x]$ for each $\gamma \in \mathbb{Z}$. Thus, the problem is reduced to counting the number of ideals of $\mathbb{Z}[x]/x(x-\alpha )(x-\beta )\mathbb{Z}[x]$ with given finite index where $0,\alpha$ and $\beta$ are distinct integers.