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

TL;DR

The authors develop a field-change framework for the Cassels–Tate pairing by constructing an Ind_{K/F} functor between SMod_K and SMod_F and proving that the Cassels–Tate pairing is preserved under this restriction of scalars. This base-change compatibility yields adjunctions between corestriction and restriction and enables transferring structural identities of Selmer groups and pairings from K to F (and vice versa). They then apply the framework to class groups, showing that reciprocity pairings on class groups with many roots of unity arise as Cassels–Tate pairings and proving symmetry phenomena in 2-adic and odd ℓ settings. The paper also develops bilinearly enhanced class groups via theta groups, and discusses decompositions of class groups in light of Cohen–Lenstra–Martinet heuristics, including several examples such as involution spins and Kramer’s norm group. Collectively, the work broadens the toolkit for translating Selmer/CT theory across fields and provides new insights into the arithmetic of class groups through a CT-lens.

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.

Figures (2)

• Figure A.1: Four group change homomorphisms
• Figure A.2: Some useful natural homomorphisms

Theorems & Definitions (63)

• Theorem 1.1
• Proposition 1.3
• Proposition 1.4
• Proposition 1.5
• Theorem 2.1: MS21, Theorem 1.3
• Definition 2.2: MS21, Definition 3.2
• Proposition 3.2
• proof : Proof of Proposition \ref{['prop:independence_of_embeddings']}
• Definition 4.1
• Definition 4.2