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 $\Gamma \subset \mathrm{End}(G)$ be a subset of the endomorphism group of $G$. A sequence $S=g_1\cdot \dots \cdot g_{\ell }$ over $G$ is a ($\Gamma$-)weighted zero-sum sequence if there are ${\gamma }_1,\dots ,{\gamma }_{\ell }\in \Gamma$ such that ${\gamma }_1(g_1)+\dots +{\gamma }_{\ell }(g_{\ell })=0$. 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.

To appear in Acta Math. Hungar