On fine Selmer groups and the greatest common divisor of signed and chromatic

Let E / Q E/\mathbb {Q} be an elliptic curve and alttext="p"> p encoding="application/x-tex">p odd prime where E"> encoding="application/x-tex">E has good supersingular reduction. F 1"> F 1 encoding="application/x-tex">F_1 denote the characteristic power series of Pontryagin dual fine Selmer group over cyclotomic alttext="double-struck Z Subscript p"> mathvariant="double-struck">Z encoding="application/x-tex">\mathbb {Z}_p -extension let 2"> 2 encoding="application/x-tex">F_2 greatest common divisor Pollack’s plus minus -adic L"> L encoding="application/x-tex">L -functions or Sprung’s sharp flat attached to , depending on whether alttext="a p Baseline left-parenthesis right-parenthesis equals 0"> a ( stretchy="false">) = 0 encoding="application/x-tex">a_p(E)=0 not-equals ≠ encoding="application/x-tex">a_p(E)\ne 0 . We study a link between divisors in Iwasawa algebra. This gives new insights into problems posed by Greenberg Pollack–Kurihara these elements.