On the plus and the minus Selmer groups for elliptic curves at supersingular primes

TL;DR

This paper generalizes the construction of plus and minus Selmer groups for elliptic curves with supersingular reduction at primes above p, and extends results on their duals in Iwasawa theory.

Contribution

It introduces a more general framework for plus and minus Selmer groups for supersingular elliptic curves and generalizes Kim's results on their dual modules.

Findings

Constructed plus and minus Selmer groups in a broader setting.

Generalized Kim's triviality results for finite submodules.

Extended Iwasawa-theoretic properties of Selmer groups.

Abstract

Let $p$ be an odd prime number, $E$ an elliptic curve defined over a number field. Suppose that $E$ has good reduction at any prime lying above $p$, and has supersingular reduction at some prime lying above $p$. In this paper, we construct the plus and the minus Selmer groups of $E$ over the cyclotomic $\mathbb Z_p$-extension in a more general setting than that of B.D. Kim, and give a generalization of a result of B.D. Kim on the triviality of finite $\Lambda$-submodules of the Pontryagin duals of the plus and the minus Selmer groups, where $\Lambda$ is the Iwasawa algebra of the Galois group of the $\mathbb Z_p$-extension.