Vol. 9, No. 10, 2015

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Bounds for Serre's open image theorem for elliptic curves over number fields

Davide Lombardo

Vol. 9 (2015), No. 10, 2347–2395
Abstract

For an elliptic curve EK without potential complex multiplication we bound the index of the image of Gal(K¯K) in GL2( ̂), the representation being given by the action on the Tate modules of E at the various primes. The bound is explicit and only depends on [K : ] and on the stable Faltings height of E. We also prove a result relating the structure of closed subgroups of GL2() to certain Lie algebras naturally attached to them.

Keywords
Galois representations, elliptic curves, Lie algebras, open image theorem
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 11F80, 14K15
Milestones
Received: 7 May 2015
Revised: 1 September 2015
Accepted: 6 October 2015
Published: 16 December 2015
Authors
Davide Lombardo
Département de Mathématiques Bâtiment 425
Université Paris-Sud 11
Faculté des Sciences d’Orsay
91405 Orsay Cedex
France