Ranks of Selmer groups in an analytic family

TL;DR

This paper investigates how the dimension of Bloch-Kato Selmer groups varies within certain p-adic Galois representations, establishing lower semi-continuity and supporting Bloch-Kato conjectures for modular forms.

Contribution

It demonstrates the lower semi-continuity of Selmer group dimensions in refined families of p-adic Galois representations, enabling new bounds and insights into Bloch-Kato conjectures.

Findings

Selmer group dimension varies lower semi-continuously in families.

Established lower bounds for Selmer groups based on continuity.

Supported Bloch-Kato conjecture predictions for modular forms.

Abstract

We study the variation of the dimension of the Bloch-Kato Selmer group of a p-adic Galois representation of a number field that varies in a refined family. We show that, if one restricts ourselves to representations that are, at every place dividing $p$, crystalline, non-critically refined, and with a fixed number of non-negative Hodge-Tate weights, then the dimension of the Selmer group varies essentially lower-semi-continuously. This allows to prove lower bounds for Selmer groups "by continuity", in particular to prove some predictions of the conjecture of Bloch-Kato for modular forms.