On the semigroup ring of holomorphic Artin L-functions
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2018
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910427410071552 |
|---|---|
| author | Cimpoeas, Mircea |
| author_facet | Cimpoeas, Mircea |
| contents | 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 $s_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_08813 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On the semigroup ring of holomorphic Artin L-functions Cimpoeas, Mircea Number Theory 11R42, 16S36 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 $s_0$. |
| title | On the semigroup ring of holomorphic Artin L-functions |
| topic | Number Theory 11R42, 16S36 |
| url | https://arxiv.org/abs/1810.08813 |