On the signed Selmer groups of congruent elliptic curves with semistable reduction at all primes above $p$

TL;DR

This paper compares signed Selmer groups of congruent elliptic curves with semistable reduction at primes above p, establishing results on Iwasawa invariants and implications for the p-parity conjecture.

Contribution

It introduces a comparison of Iwasawa $\lambda$-invariants for signed Selmer groups of congruent elliptic curves and extends parity results to these curves.

Findings

Comparison formula for Iwasawa $\lambda$-invariants of congruent elliptic curves.

If the p-parity conjecture holds for one curve, it holds for the other.

Generalization of Hatley's observation on the parity of signed Selmer groups.

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 comparsion 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 elliptic curve. In the midst of proving this latter result, we also generalize an observation of Hatley on the parity of the signed Selmer groups.