On the semigroup ring of holomorphic Artin L-functions

Abstract

Let $K/\mathbb Q$ be a finite Galois extension and let $χ_1,\ldots,χ_r$ be the irreducible characters of the Galois group $G:=Gal(K/\mathbb Q)$. Let $f_1:=L(s,χ_1),\ldots,f_r:=L(s,χ_r)$ be their associated Artin L-functions. For $s_0\in \mathbb C\setminus\{1\}$, we denote $Hol(s_0)$ the semigroup of Artin $L$-functions, holomorphic at $s_0$. Let $\mathbb F$ be a field with $\mathbb C \subseteq \mathbb F \subseteq \mathcal M_{<1}:=$ the field of meromorphic functions of order $<1$. We note that the semigroup ring $\mathbb F[Hol(s_0)]$ 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 $I_{H(s_0)}=(0)$. 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 $s_0$.

Theorems & Definitions (28)

• Proposition 1.1
• proof
• Remark 1.2
• Proposition 1.3
• proof
• Theorem 1.4
• proof
• Remark 1.5
• Theorem 1.6
• proof