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 \(\Gamma \subset {\rm End} (G)\) 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.