A formula of Perrin-Riou and characteristic power series of signed Selmer groups

TL;DR

This paper proves a conjecture relating the leading term of a characteristic power series of signed Selmer groups for supersingular elliptic curves to a formula by Perrin-Riou, advancing understanding in $p$-adic number theory.

Contribution

It establishes a new formula connecting the characteristic power series of signed Selmer groups to Perrin-Riou's work, confirming a conjecture in the context of supersingular primes.

Findings

Proved a conjecture relating Selmer groups and $p$-adic $L$-functions.

Derived a formula for the leading term of the characteristic power series.

Connected arithmetic $p$-adic $L$-functions with Perrin-Riou's modules.

Abstract

We prove a conjecture of Kundu--Ray, following from the $p$-adic Birch--Swinnerton-Dyer conjecture for supersingular primes by Bernardi--Perrin-Riou and Kato's Main Conjecture, predicting an expression for the leading term (up to a $p$-adic unit) of a characteristic power series of Kobayashi's signed Selmer groups attached to elliptic curves $E/\mathbb{Q}$ with supersingular reduction at a prime $p>2$ with $a_p=0$. The proof is deduced from a similar formula due to Perrin-Riou for a generator of her module of arithmetic $p$-adic $L$-functions with values in the Dieudonn\'{e} module of $E$.