On Dirichlet Series of a Certain Commutative Matrix Ring
Hisaichi Midorikawa · Tokyo Journal of Mathematics · 1997
To Professor Takeshi Hirai on his 60th birthday 1. Introduction.Let $GL(n, Z)$ denote the modular group of degree $n$ over the ring of integers Z.For a regular element $\zeta$ in $GL(n, Z)$ , let $R$ denote the ring generated by $\zeta$ over $Z$ and let $f(X)$ be its characteristic polynomial.The purpose of this paper is to show that a special value of certain Dirichlet series $\zeta_{R}(s)$ of $R$ at $s=1$ gives rise to an ideal regulator-class number formula for $R$ , which is a generalization of the classical regulator-class number formula for the Dedekind zeta functions of a number field.Before stating our results, we need some preparation.An ideal $\mathfrak{a}\subset R$ is said to be nonsingular if the index $(R:\mathfrak{a})$ (as group) is finite, in which case the norm of $\mathfrak{a},$ $N\mathfrak{a}$ , is defined to be this index.Let $Q[\zeta]$ be the ring generated by $\zeta$ over the field of rationalsWe shall prove (Lemma 3.6 below) that the index $(E_{O} : E_{\mathfrak{a}})$ is finite.Let us define the ideal class semigroup $G$ of the ring $R$ .Two fractional ideals $\mathfrak{a}$ and $b$ are said to be equivalent if there exists an invertible element $\lambda$ in $Q [\zeta]$ such that $\lambda \mathfrak{a}=b$ .We denote by $G$ the set of all equivalence classes and a class in $G$ by $C=C(\mathfrak{a})$ with a representative $\mathfrak{a}$ .Note that $G$ is a semigroup under the canonical multiplication.