On characteristic power series of dual signed Selmer groups

TL;DR

This paper establishes a relationship between the size of the $p$-primary Bloch-Kato Selmer group for modular forms at non-ordinary primes and the constant term of characteristic power series of signed Selmer groups over cyclotomic extensions, generalizing previous results.

Contribution

It extends the connection between Selmer groups and characteristic power series to non-ordinary primes, broadening the scope of earlier ordinary case results.

Findings

Relates $p$-primary Selmer group size to characteristic power series

Generalizes Vigni and Longo's results to non-ordinary primes

Builds on prior work for elliptic curves in ordinary and supersingular cases

Abstract

We relate the cardinality of the $p$-primary part of the Bloch-Kato Selmer group over $\mathbb{Q}$ attached to a modular form at a non-ordinary prime $p$ to the constant term of the characteristic power series of the signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. This generalizes a result of Vigni and Longo in the ordinary case. In the case of elliptic curves, such results follow from earlier works by Greenberg, Kim, the second author, and Ahmed-Lim, covering both the ordinary and most of the supersingular case.