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arXiv · 2103.06150

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

Antonio Lei·R. Sujatha
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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.

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Antonio Lei, R. Sujatha. 2021-03-10. On fine Selmer groups and the greatest common divisor of signed and chromatic $p$-adic $L$-functions. https://doi.org/10.1090/proc/15480

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