On the semigroup ring of holomorphic Artin L-functions

Abstract

Let $K/\mathbb{Q}$ be a finite Galois extension and let ${\chi }_1,\dots ,{\chi }_r$ be the irreducible characters of the Galois group $G:=Gal(K/\mathbb{Q})$. Let $f_1:=L(s,{\chi }_1),\dots ,f_r:=L(s,{\chi }_r)$ 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 $\mathbb{F}[H(s_0)]\subseteq \mathbb{F}[x_1,\dots ,x_r]$, 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 $f_1,\dots ,f_r$ have only simple zeros and simple poles at $s_0$.

12 pages, some improvements over the first version (Theorem 1.6)

References in corpus (2)

• Independence of Artin L-functions
• On the Semigroup of Artin's L-functions holomorphic at $s_0$. II

Cited by in corpus (1)

• Two semigroup rings associated to a finite set of germs of meromorphic functions