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

Abstract

Let E / Q E/\mathbb {Q} be an elliptic curve and p p an odd prime where E E has good supersingular reduction. Let F 1 F_1 denote the characteristic power series of the Pontryagin dual of the fine Selmer group of E E over the cyclotomic Z p \mathbb {Z}_p -extension of Q \mathbb {Q} and let F 2 F_2 denote the greatest common divisor of Pollack’s plus and minus p p -adic L L -functions or Sprung’s sharp and flat p p -adic L L -functions attached to E E , depending on whether a p ( E ) = 0 a_p(E)=0 or a p ( E ) ≠ 0 a_p(E)\ne 0 . We study a link between the divisors of F 1 F_1 and F 2 F_2 in the Iwasawa algebra. This gives new insights into problems posed by Greenberg and Pollack–Kurihara on these elements.