Elasticity of Orders from the $S$-relative Davenport Constant: an Arithmetic Application of a Number-Theoretic Investigation

TL;DR

This paper introduces the $S$-relative Davenport constant to analyze the elasticity of non-maximal orders in algebraic number fields, extending previous work to new classes of orders and providing conditions for elasticity equivalence.

Contribution

The authors develop the $S$-relative Davenport constant as a novel tool to study elasticity in non-integrally closed orders, expanding its application beyond quadratic forms.

Findings

Determined elasticity for orders with prime conductor ideals.

Identified conditions for orders to share elasticity with their maximal orders.

Extended the applicability of the $S$-relative Davenport constant to quadratic number fields.

Abstract

Orders in algebraic number fields have long been objects of central interest in algebraic number theory. Despite non-maximal orders failing to be Dedekind, the present authors have previously shown that the structure of the ideal class group may still contain enough information to determine elasticity. In this paper, we develop the $S$-relative Davenport constant, which builds on previous work by M. Ska{\l}ba. Although Ska{\l}ba's original construction was defined to aid in the study of binary quadratic forms, we show that this related invariant is the exact tool needed to tackle the question of elasticity in non-integrally closed orders. In particular, we investigate the elasticity of orders $\mathcal{O}$ whose conductor ideal $I=(\mathcal{O}:\mathcal{O}_K)$ is prime as an ideal of $\mathcal{O}$, as well as orders in quadratic number fields with primary conductor. We also give conditions under which $\mathcal{O}$ will have the same elasticity as the full ring of integers $\mathcal{O}_K$.