PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On the semigroup ring of holomorphic Artin $L$-functions

Volume 160 / 2020

Mircea Cimpoeaş Colloquium Mathematicum 160 (2020), 283-295 MSC: Primary 11R42; Secondary 16S36. DOI: 10.4064/cm7750-3-2019 Published online: 17 February 2020


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 $s_0$.


  • Mircea CimpoeaşSimion Stoilow Institute of Mathematics
    Research unit 5
    P.O. Box 1-764
    Bucureşti 014700, Romania
    Department of Mathematical Methods and Models
    Faculty of Applied Sciences
    Politehnica University of Bucharest
    Bucureşti 060042, Romania

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image