Field change for the Cassels-Tate pairing and applications to class groups

TL;DR

This paper explores how the Cassels-Tate pairing behaves under field extensions and applies these insights to analyze class groups in the context of Cohen-Lenstra heuristics.

Contribution

It introduces a restriction of scalars functor for Galois modules with local conditions and studies its compatibility with the Cassels-Tate pairing.

Findings

The restriction of scalars functor preserves the Cassels-Tate pairing.

Applications to class groups support Cohen-Lenstra heuristics.

Provides a new framework for understanding Galois modules across field extensions.

Abstract

In previous work, the authors defined a category $SMod_F$ of finite Galois modules decorated with local conditions for each global field $F$. In this paper, given an extension $K/F$ of global fields, we define a restriction of scalars functor from $SMod_K$ to $SMod_F$ and show that it behaves well with respect to the Cassels-Tate pairing. We apply this work to study the class groups of global fields in the context of the Cohen-Lenstra heuristics.