On Bloch-Kato Selmer groups and Iwasawa theory of $p$-adic Galois representations

TL;DR

This paper extends Greenberg's relation between Selmer groups and Iwasawa modules from elliptic curves to more general $p$-adic Galois representations, including those from $p$-ordinary modular forms.

Contribution

It generalizes Greenberg's results to a broader class of $p$-adic Galois representations, enhancing understanding of their Iwasawa theory.

Findings

Extended Greenberg's relation to general $p$-adic Galois representations.

Included representations attached to $p$-ordinary modular forms.

Provided new insights into Selmer groups over cyclotomic extensions.

Abstract

A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic $\mathbb Z_p$-extensions at good ordinary primes $p$. We extend Greenberg's result to more general $p$-adic Galois representations, including a large subclass of those attached to $p$-ordinary modular forms of level $\Gamma_0(N)$ with $p\nmid N$.