arXiv subjects

Jared Kettinger

Publications and source records attributed to Jared Kettinger.

5 recordsLinked to original sources

arXiv2026

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. 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 $\mathcal{O}_K$.

math.AC↗

arXiv2025

Galois Action and Localization in Number Fields

For a Galois number field $K$, the Galois group $\text{Gal}(K/\mathbb{Q})$ acts on the class group $\text{Cl}_K$ in a very natural way: $\sigma\cdot[I]=[\sigma(I)]$ for any $\sigma \in \text{Gal}(K/\mathbb{Q})$, $[I]\in \text{Cl}_K$. In this paper, we will explore how the unique properties of this group action work together to elucidate the relationship between these two groups -- developing and expanding upon some known results from a new perspective. To this end, we explore the class groups of localizations of the ring of integers $\mathcal{O}_K$. These turn out to be powerful tools for understanding $\text{Cl}_K$ and overrings of $\mathcal{O}_K$. The paper concludes with some interesting observations about normset arithmetic and complexity -- topics intimately related to this action.

math.NT↗

arXiv2025

Automorphically Equivalent Elements of Finite Abelian Groups

Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. This result leads to our development of the two fastest known algorithms to determine if two elements of a finite abelian group are automorphic images of one another. The second algorithm also computes $G/\langle x \rangle$ in a near-linear time algorithm for groups, most feasible when the group has exponent at most $10^{20}$. We conculde with an algorithm that computes the automorphic orbits of finite abelian groups.

math.GR↗

arXiv2025

A Generalized Davenport Constant of the Second Kind

In this paper, we explore a ring invariant which is closely related to the Davenport constant of a group. In particular, we will calculate this invariant for a certain class of rings of integers and their orders and use it to understand factorization properties of the latter. To this end, we also examine the well-behaved class of Galois-invariant orders.

math.AC↗

arXiv2025

Elasticity of Orders with Prime Conductor

Let $R$ be an order in a number field whose conductor ideal $P := (R:\overline{R})$ is prime in the ring of integers $\overline{R}$. In this paper, we explore the factorization properties of such orders. Most notably, we give a complete characterization of the elasticity of $R$ in terms of its class group. We conclude with an application to the computation of class groups of certain orders.

math.AC↗