Genus character $L$-functions of quadratic orders in an adelic way and maximal orders of matrix algebras

Abstract

For a quadratic extension $K$ of ${\mathbb Q}$, we consider orders $O$ in $K$ that are not necessarily maximal and the ideal class group $Cl^+(O)$ in the narrow sense of proper ideals of $O$. Characters of $Cl^+(O)$ of order at most two are traditionally called genus characters. Explicit description of such characters is known classically, but explicit $L$-functions associated to those characters are only recently obtained partially by Chinta and Offen and completely by Kaneko and Mizuno. As remarked in the latter paper, the present author also obtained the formula of such $L$-functions independently. Indeed, here we will give a simple and transparent alternative proof of the formula by rewriting explicit genus characters and their values in an adelic way starting from scratch. We also add an explicit formula for the genus number in the wide sense, which is maybe known but rarely treated. As an appendix we give an ideal-theoretic characterization of isomorphism classes of maximal orders of the matrix algebras $M_n(F)$ over a number field $F$ up to $GL_n(F)$ and $GL_n^+(F)$ conjugation respectively, and apply genus numbers to count them when $n=2$ and $F$ is quadratic. To avoid any misconception, we include some easy known details.