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Antonio Lei; Sarah Livia Zerbes
Signed Selmer groups over $p$-adic Lie extensions
Journal de théorie des nombres de Bordeaux, 24 no. 2 (2012), p. 377-403, doi: 10.5802/jtnb.802
Article PDF | Reviews MR 2950698 | Zbl 1283.11154

Résumé - 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 Kobayashi’s conditions via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case.

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