English
The symplectic type of congruences between elliptic curves
Post-edited
Auteurs : Cremona, John (Auteur de la Conférence)
CIRM (Editeur )
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
CIRM (Editeur )
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 : This is joint work with Nuno Freitas (Warwick).
Keywords : elliptic curves; Galois representations
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
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 04/07/2019 Date de captation : 10/06/2019 Sous collection : Research talks arXiv category : Number Theory Domaine : Number Theory Format : MP4 (.mp4) - HD Durée : 00:55:05 Audience : Researchers Download : https://videos.cirm-math.fr/2019-06-10_Cremona.mp4 
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Informations sur la Rencontre
Nom de la rencontre : AGCT - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT - Arithmétique, géométrie, cryptographie et théorie des codesOrganisateurs 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
Données de citation
DOI : 10.24350/CIRM.V.19537703Citer cette vidéo: 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 |
Voir aussi
- [Multi angle] On the work and persona of Gilles Lachaud / Auteur de la Conférence Ghorpade, Sudhir.
- [Multi angle] Isolated points on modular curves / Auteur de la Conférence Viray, Bianca.
- [Multi angle] Gilles Lachaud: friend and mathematician / Auteur de la Conférence Tsfasman, Michael.
- [Multi angle] Ore polynomials and application to coding theory / Auteur de la Conférence Caruso, Xavier.
- [Multi angle] The coset leader weight enumerator of the code of the twisted cubic / Auteur de la Conférence Pellikaan, Ruud.
- [Multi angle] Good recursive towers / Auteur de la Conférence Bassa, Alp.
Bibliographie
- FISHER, Tom. On families of 7-and 11-congruent elliptic curves. LMS Journal of Computation and Mathematics, 2014, vol. 17, no 1, p. 536-564. - https://doi.org/10.1112/S1461157014000059
- FREITAS, Nuno et KRAUS, Alain. On the symplectic type of isomorphims of the p-torsion of elliptic curves. arXiv preprint arXiv:1607.01218, 2016. - https://arxiv.org/abs/1607.01218
- HALBERSTADT, Emmanuel et KRAUS, Alain. Sur la courbe modulaire $X_E(7)$. Experimental mathematics, 2003, vol. 12, no 1, p. 27-40. - https://projecteuclid.org/euclid.em/1064858782
- The LMFDB Collaboration, The L-functions and Modular Forms Database. - http://www.lmfdb.org.
- POONEN, Bjorn, SCHAEFER, Edward F., STOLL, Michael, et al. Twists of $ X (7) $ and primitive solutions to $ x^ 2+ y^ 3= z^ 7$. Duke Mathematical Journal, 2007, vol. 137, no 1, p. 103-158. - https://projecteuclid.org/euclid.dmj/1173373452
