On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms

Abstract

Let $G$ be an additive finite abelian group and $\Gamma \subset \operatorname{End} (G)$ be a subset of the endomorphism group of $G$. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ over $G$ is a ($\Gamma$-)weighted zero-sum sequence if there are $\gamma_1, \ldots, \gamma_{\ell} \in \Gamma$ such that $\gamma_1 (g_1) + \ldots + \gamma_{\ell} (g_{\ell})=0$. We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group $\Gamma$) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.