On the signed Selmer groups of congruent elliptic curves with semistable reduction at all primes above $p$
Volume 197 / 2021
Acta Arithmetica 197 (2021), 353-377
MSC: Primary 11R23; Secondary 11G05, 11S25.
DOI: 10.4064/aa190711-21-6
Published online: 14 December 2020
Abstract
Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $\lambda $-invariants of these signed Selmer groups for two congruent elliptic curves over the cyclotomic $\mathbb{Z}_{p} $-extension in the spirit of Greenberg–Vatsal and B. D. Kim. As an application of our comparison formula, we show that if the $p$-parity conjecture is true for one of the congruent elliptic curves, then it is also true for the other. While proving this last result, we also generalize an observation of Hatley on the parity of signed Selmer groups.
