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H 2 The symplectic type of congruences between elliptic curves

Auteurs : Cremona, John (Auteur de la Conférence)
CIRM (Editeur )

 Loading the player... elliptic curves and Galois representations congruences - symplectic types the Frey-Mazur conjecture congruences in the LMFDB finding congruences via sieving determining the symplectic type Frey-Mazur for the database twists congruences between twists

Résumé : In this talk I will describe a systematic investigation into congruences between the mod $p$ torsion modules of elliptic curves defined over $\mathbb{Q}$. For each such curve $E$ and prime $p$ the $p$-torsion $E[p]$ of $E$, is a 2-dimensional vector space over $\mathbb{F}_{p}$ which carries a Galois action of the absolute Galois group $G_{\mathbb{Q}}$. The structure of this $G_{\mathbb{Q}}$-module is very well understood, thanks to the work of J.-P. Serre and others. When we say the two curves $E$ and $E'$ are ”congruent” we mean that $E[p]$ and $E'[p]$ are isomorphic as $G_{\mathbb{Q}}$-modules. While such congruences are known to exist for all primes up to 17, the Frey-Mazur conjecture states that p is bounded: more precisely, that there exists $B$ > 0 such that if $p > B$ and $E[p]$ and $E'[p]$ are isomorphic then $E$ and $E'$ are isogenous. We report on work toward establishing such a bound for the elliptic curves in the LMFDB database. Secondly, we describe methods for determining whether or not a given isomorphism between $E[p]$ and $E'[p]$ is symplectic (preserves the Weil pairing) or antisymplectic, and report on the results of applying these methods to the curves in the database.
This is joint work with Nuno Freitas (Warwick).

Keywords : elliptic curves ; Galois representations

Codes MSC :
11A07 - Congruences; primitive roots; residue systems
11G05 - Elliptic curves over global fields
14H52 - Elliptic curves

Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/1921/Slides/Cremona.pdf

 Informations sur la Vidéo Langue : Anglais Date de publication : 04/07/2019 Date de captation : 10/06/2019 Collection : Research talks Format : MP4 (.mp4) - HD Durée : 00:55:05 Domaine : Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2019-06-10_Cremona.mp4 Informations sur la rencontre Nom de la rencontre : Arithmetic, Geometry, Cryptography and Coding Theory / Arithmétique, géométrie, cryptographie et théorie des codesOrganisateurs de la rencontre : Ballet, Stéphane ; Bisson, Gaetan ; Bouw, IreneDates : 10/06/2019 - 14/06/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/1921.html Citation Data DOI : 10.24350/CIRM.V.19537703 Cite this video as: Cremona, John (2019). The symplectic type of congruences between elliptic curves. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19537703 URI : http://dx.doi.org/10.24350/CIRM.V.19537703

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