THE NUMBER OF IDEALS OF Z[x] CONTAINING x(x− α)(x− β) WITH GIVEN INDEX

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 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 ζ〈B〉(s) = ∑ n≥1 ann −s is well-defined where an is the number of ideals of 〈B〉 with index n. In this article we aim to find an explicit form of ζ〈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 Z[x]/m(x)Z[x] where m(x) is the minimal polynomial of B over Q, and Z[x]/m(x)Z[x] is isomorphic to Z[x]/m(x+γ)Z[x] for each γ ∈ Z. Thus, the problem is reduced to counting the number of ideals of Z[x]/x(x−α)(x− β)Z[x] with given finite index where 0, α and β are distinct integers.