Equidistribution of realizable Steinitz classes for Kummer extensions

Abstract: Let $\ell$ be prime, and $K$ be a number field containing the $\ell$-th roots of unity. We use techniques from classical algebraic number theory to prove that the Steinitz classes of $\Z/\ell\Z$ extensions of $K$ are equidistributed among realizable classes in the ideal class group of $K$. Similar equidistribution results have been proved for Galois groups $S_2$ and $S_3$ by Kable and Wright and $S_4$ and $S_5$ by Bhargava, Shankar, and Wang using the theory of prehomogeneous vector spaces, but this is the first complete equidistribution result for an infinite class of Galois groups.

Next, we discuss generalizations of this result to elementary-$\ell$ Galois groups using $V_4$ as an example. Additionally, we will give some initial results for Steinitz classes of ray class fields.

number theory

Audience: researchers in the topic

( paper )

Five College Number Theory Seminar