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

Lombardo, Davide

doi:10.2140/ant.2015.9.2347

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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

DOI: 10.2140/ant.2015.9.2347

Abstract

For an elliptic curve $E ∕ K$ without potential complex multiplication we bound the index of the image of Gal $(\bar{K} ∕ K)$ in GL $_{2} (\hat{ℤ})$ , 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 $[K : ℚ]$ and on the stable Faltings height of $E$ . We also prove a result relating the structure of closed subgroups of GL $_{2} (ℤ_{ℓ})$ 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

Vol. 9, No. 10, 2015