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

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 $a_p(E)\ne 0$. 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.

To appear in Proceedings of the AMS

References in corpus (1)

• The Iwasawa Main Conjecture for elliptic curves at odd supersingular primes