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H 1 Computing the image of Galois representations attached to elliptic curves

Auteurs : Sutherland, Andrew (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Let $E$ be an elliptic curve over a number field $K$. For each integer $n > 1$ the action of the absolute Galois group $G_K := Gal(\overline{K}/K)$ on the $n$-torsion subgroup $E [n]$ induces a Galois representation $\rho_{E,n}:G_K \rightarrow$ Aut$(E[n]) \backsimeq GL_2(\mathbb{Z} /n\mathbb{Z})$. The representations $\rho_{E,n}$ form a compatible system, and after taking inverse limits one obtains an adelic representation $\rho_E:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$. If $E/K$ does not have $CM$, then Serre’s open image theorem implies that the image of $\rho_E$ has finite index in $GL_2(\hat{\mathbb{Z}})$; in particular, $\rho_{E,\ell}$ is surjective for all but finitely many primes $\ell$.
I will present an algorithm that, given an elliptic curve $E/K$ without $CM$, determines the image of $\rho_{E,\ell}$ in $GL_2(\mathbb{Z} /\ell\mathbb{Z})$ up to local conjugacy for every prime $\ell$ for which $\rho_{E,\ell}$ is non-surjective. Assuming the generalized Riemann hypothesis, the algorithm runs in time that is polynomial in the bit-size of the coefficients of an integral Weierstrass model for $E$. I will then describe a probabilistic algorithm that uses this information to compute the index of $\rho_E$ in $GL_2(\hat{\mathbb{Z}})$.

Codes MSC :
11G05 - Elliptic curves over global fields
11Y16 - Algorithms; complexity

 Informations sur la Vidéo Langue : Anglais Date de publication : 04/06/15 Date de captation : 18/05/15 Collection : Research talks ; Number Theory Format : quicktime ; audio/x-aac Durée : 00:28:53 Domaine : Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-05-18_Sutherland.mp4 Informations sur la rencontre Nom de la rencontre : Arithmetics, geometry, cryptography and coding theory / Arithmétique, géométrie, cryptographie et théorie des codesOrganisateurs de la rencontre : Bassa, Alp ; Couvreur, Alain ; Kohel, DavidDates : 18/05/15 - 22/05/15 Année de la rencontre : 2015 URL Congrès : http://conferences.cirm-math.fr/1193.htmlCitation Data DOI : 10.24350/CIRM.V.18764803 Cite this video as: Sutherland, Andrew (2015). Computing the image of Galois representations attached to elliptic curves. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18764803 URI : http://dx.doi.org/10.24350/CIRM.V.18764803

Bibliographie

1. Sutherland, A.V. (2015). Computing images of Galois representations attached to elliptic curves. - http://arxiv.org/abs/1504.07618

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