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

In previous work, the authors defined a category $SMo{d}_F$ of finite Galois modules decorated with local conditions for each global field $F$. In this paper, given an extension $K\mathrm{/}F$ of global fields, we define a restriction of scalars functor from $SMo{d}_K$ to $SMo{d}_F$ 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.