Signed Selmer Groups over p-adic Lie Extensions

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\ge 3$ 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 $(\varphi ,\Gamma )$-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.

21 pages

Cited by in corpus (5)

• On Selmer groups in the supersingular reduction case
• Non-commutative p-adic L-functions for supersingular primes
• On the cohomology of Kobayashi's plus/minus norm groups and applications
• Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p
• On the cohomology of plus/minus Selmer groups of supersingular elliptic curves in weakly ramified base fields