Signed Selmer Groups over p-adic Lie Extensions

TL;DR

This paper extends the theory of plus/minus Selmer groups for supersingular elliptic curves over cyclotomic extensions to more general p-adic Lie extensions, using advanced $(, )$-module techniques.

Contribution

It generalizes Kobayashi's Selmer groups to p-adic Lie extensions, employing $(, )$-modules and overconvergent power series methods.

Findings

Selmer groups are described via jumping conditions and overconvergent power series.

The approach recovers classical Selmer groups in the ordinary case.

Provides a framework for Selmer groups over broader p-adic Lie extensions.

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\geq 3$ and $a_p=0$. We generalise the definition of Kobayashi's plus/minus Selmer groups over $\mathbb{Q}(\mu_{p^\infty})$ to $p$-adic Lie extensions $K_\infty$ of $\mathbb{Q}$ containing $\mathbb{Q}(\mu_{p^\infty})$, using the theory of $(\phi,\Gamma)$-modules and Berger's comparison isomorphisms. We show that these Selmer groups can be equally described using the "jumping conditions" of Kobayashi via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case.