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

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.

Skałba.

Although Skałba'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 $\mathcal{O}_K$.