On the semigroup ring of holomorphic Artin $L$-functions
Volume 160 / 2020
Abstract
Let $K/\mathbb Q$ be a finite Galois extension and let $\chi _1,\ldots ,\chi _r$ be the irreducible characters of the Galois group $G := \operatorname {\rm Gal} (K/\mathbb Q)$. Let $f_1:=L(s,\chi _1),\ldots ,f_r:=L(s,\chi _r)$ be their associated Artin $L$-functions. For $s_0\in \mathbb C\setminus \{1\}$, we denote by $\operatorname {\rm Hol} (s_0)$ the semigroup of Artin $L$-functions, holomorphic at $s_0$. Let $\mathbb F$ be a field with $\mathbb C \subseteq \mathbb F \subseteq \mathcal M_{ \lt 1}:=$ the field of meromorphic functions of order $ \lt 1$. We note that the semigroup ring $\mathbb F[\operatorname {\rm Hol} (s_0)]$ is isomorphic to a toric ring $\mathbb F[H(s_0)]\subseteq \mathbb F[x_1,\ldots ,x_r]$, where $H(s_0)$ is an affine subsemigroup of $\mathbb N^r$ minimally generated by at least $r$ elements, and we describe $\mathbb F[H(s_0)]$ when the toric ideal $I_{H(s_0)}$ is $(0)$. Also, we describe $\mathbb F[H(s_0)]$ and $I_{H(s_0)}$ when $f_1,\ldots ,f_r$ have only simple zeros and simple poles at $s_0$.
