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

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...

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Bibliographic Details
Published in:arXiv.org 2020-06
Main Authors: Ahmed, Suman, Lim, Meng Fai
Format: Article
Language:English
Subjects:
Curves
Parity
Reduction
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1912.08430