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The symplectic type of congruences between elliptic curves
Post-edited
Authors : Cremona, John (Author of the conference)
CIRM (Publisher )
11A07 - Congruences; primitive roots; residue systems
11G05 - Elliptic curves over global fields
14H52 - Elliptic curves
Additional resources :
https://www.cirm-math.fr/RepOrga/1921/Slides/Cremona.pdf
CIRM (Publisher )
Abstract :
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
MSC Codes : 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
Additional resources :
https://www.cirm-math.fr/RepOrga/1921/Slides/Cremona.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 04/07/2019 Conference Date : 10/06/2019 Subseries : Research talks arXiv category : Number Theory Mathematical Area(s) : Number Theory Format : MP4 (.mp4) - HD Video Time : 00:55:05 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-06-10_Cremona.mp4 
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Information on the Event
Event Title : AGCT - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT - Arithmétique, géométrie, cryptographie et théorie des codesEvent Organizers : Ballet, Stéphane ; Bisson, Gaetan ; Bouw, Irene Dates : 10/06/2019 - 14/06/2019 Event Year : 2019 Event URL : https://conferences.cirm-math.fr/1921.html
Citation Data
DOI : 10.24350/CIRM.V.19537703Cite 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 |
See Also
- [Multi angle] On the work and persona of Gilles Lachaud / Author of the conference Ghorpade, Sudhir.
- [Multi angle] Isolated points on modular curves / Author of the conference Viray, Bianca.
- [Multi angle] Gilles Lachaud: friend and mathematician / Author of the conference Tsfasman, Michael.
- [Multi angle] Ore polynomials and application to coding theory / Author of the conference Caruso, Xavier.
- [Multi angle] The coset leader weight enumerator of the code of the twisted cubic / Author of the conference Pellikaan, Ruud.
- [Multi angle] Good recursive towers / Author of the conference Bassa, Alp.
Bibliography
- 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
