On fine Selmer groups and the greatest common divisor of signed and chromatic $p$-adic $L$-functions

TL;DR

This paper investigates the relationship between the fine Selmer group's characteristic power series and the greatest common divisor of signed and chromatic $p$-adic $L$-functions for elliptic curves with supersingular reduction, offering new insights into Iwasawa theory.

Contribution

It establishes a link between the divisors of the fine Selmer group's characteristic series and the gcd of signed and chromatic $p$-adic $L$-functions, advancing understanding in Iwasawa theory.

Findings

Identifies a connection between divisors of $F_1$ and $F_2$ in the Iwasawa algebra.

Provides new perspectives on Greenberg's conjectures.

Addresses problems posed by Pollack and Kurihara.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ has good supersingular reduction. Let $F_1$ denote the characteristic power series of the Pontryagin dual of the fine Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ and let $F_2$ denote the greatest common divisor of Pollack's plus and minus $p$-adic $L$-functions or Sprung's sharp and flat $p$-adic $L$-functions attached to $E$, depending on whether $a_p(E)=0$ or $a_p(E)\ne0$. We study a link between the divisors of $F_1$ and $F_2$ in the Iwasawa algebra. This gives new insights into problems posed by Greenberg and Pollack--Kurihara on these elements.