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Mathematics > Number Theory

arXiv:1403.3813 (math)
[Submitted on 15 Mar 2014 (v1), last revised 8 Oct 2015 (this version, v5)]

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

Authors:Davide Lombardo
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Abstract:For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\operatorname{Gal}(\bar{K}/K)$ in $\operatorname{GL}_2(\hat{\mathbb{Z}})$, the representation being given by the action on the Tate modules of $E$ at the various primes. The bound is effective and only depends on $[K:\mathbb{Q}]$ and on the stable Faltings height of $E$. We also prove a result relating the structure of subgroups of $\operatorname{GL}_2(\mathbb{Z}_\ell)$ to certain Lie algebras naturally attached to them.
Comments: Final version, accepted for publication in Algebra and Number Theory
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
MSC classes: 11G05, 11F80
Cite as: arXiv:1403.3813 [math.NT]
  (or arXiv:1403.3813v5 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1403.3813

Submission history

From: Davide Lombardo [view email]
[v1] Sat, 15 Mar 2014 14:49:15 UTC (33 KB)
[v2] Wed, 16 Apr 2014 11:57:59 UTC (44 KB)
[v3] Fri, 22 May 2015 07:39:35 UTC (43 KB)
[v4] Wed, 3 Jun 2015 00:11:33 UTC (44 KB)
[v5] Thu, 8 Oct 2015 18:04:41 UTC (45 KB)
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