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arXiv · 2506.12999

Equidistribution of realizable Steinitz classes for cyclic Kummer extensions

Brody Lynch
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Abstract

Let $\ell$ be prime, and $K$ be a number field containing the $\ell$-th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of $\mathbb Z/\ell\mathbb Z$ extensions of $K$ ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of $K$. For $\ell = 2$, this was proved by Kable and Wright using the deep theory of prehomogeneous vector spaces. Foster proved that Steinitz classes are uniformly distributed between realizable classes for tamely ramified elementary-$m$ extensions using the theory of Galois modules; our approach eliminates this tameness hypothesis.

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BibTeXRIS

Brody Lynch. 2025-06-15. Equidistribution of realizable Steinitz classes for cyclic Kummer extensions. https://doi.org/10.1016/j.jnt.2025.09.022

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