On characteristic power series of dual signed Selmer groups

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.