Signed Selmer groups over p -adic Lie extensions

Antonio Lei[1]; Sarah Livia Zerbes[2]

  • [1] School of Mathematical Sciences Monash University Clayton, VIC 3800 Australia Since December 2011 : Department of Mathematics and Statistics Burnside Hall McGill University Montreal QC Canada H3A 2K6
  • [2] Department of Mathematics Harrison Building University of Exeter Exeter EX4 4QF, UK

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 2, page 377-403
  • ISSN: 1246-7405

Abstract

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Let E be an elliptic curve over ℚ with good supersingular reduction at a prime p ≥ 3 and a p = 0 . We generalise the definition of Kobayashi’s plus/minus Selmer groups over ℚ ( μ p ∞ ) to p -adic Lie extensions K ∞ of ℚ containing ℚ ( μ p ∞ ) , using the theory of ( ϕ , Γ ) -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.

How to cite

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Lei, Antonio, and Zerbes, Sarah Livia. "Signed Selmer groups over $p$-adic Lie extensions." Journal de Théorie des Nombres de Bordeaux 24.2 (2012): 377-403. <http://eudml.org/doc/251038>.

@article{Lei2012,
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.},
affiliation = {School of Mathematical Sciences Monash University Clayton, VIC 3800 Australia Since December 2011 : Department of Mathematics and Statistics Burnside Hall McGill University Montreal QC Canada H3A 2K6; Department of Mathematics Harrison Building University of Exeter Exeter EX4 4QF, UK},
author = {Lei, Antonio, Zerbes, Sarah Livia},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Selmer groups, –adic Lie extensions; elliptic curves; good reduction; supersingular reduction},
language = {eng},
month = {6},
number = {2},
pages = {377-403},
publisher = {Société Arithmétique de Bordeaux},
title = {Signed Selmer groups over $p$-adic Lie extensions},
url = {http://eudml.org/doc/251038},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Lei, Antonio
AU - Zerbes, Sarah Livia
TI - Signed Selmer groups over $p$-adic Lie extensions
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/6//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 2
SP - 377
EP - 403
AB - 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.
LA - eng
KW - Selmer groups, –adic Lie extensions; elliptic curves; good reduction; supersingular reduction
UR - http://eudml.org/doc/251038
ER -

References

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