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H 2 The symplectic type of congruences between elliptic curves

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

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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 codes
Organisateurs de la rencontre : Ballet, Stéphane ; Bisson, Gaetan ; Bouw, Irene
Dates : 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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