On the semigroup ring of holomorphic Artin L-functions

Abstract

Let K/Q be a finite Galois extension and let χ 1 , . . . , χ r be the irreducible characters of the Galois group G := Gal(K/Q). Let f 1 := L(s, χ 1 ), . . . , f r := L(s, χ r ) be their associated Artin L-functions. For s 0 ∈ C \ {1}, we denote Hol(s 0 ) the semigroup of Artin L-functions, holomorphic at s 0 . Let F be a field with C ⊆ F ⊆ M <1 := the field of meromorphic functions of order < 1. We note that the semigroup ring, where H(s 0 ) is an affine subsemigroup of N r minimally generated by at least r elements, and we describe F[H(s 0 )] when the toric ideal I H(s 0 ) = (0). Also, we describe F[H(s 0 )] and I H(s 0 ) when f 1 , . . . , f r have only simple zeros and simple poles at s 0 .