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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). 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 ...

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Documents 11G05 | enregistrements trouvés : 81

The symplectic type of congruences between elliptic curves Cremona, John | CIRM H Nouveau

Stable models for modular curves in prime level Parent, Pierre | CIRM H Nouveau

Integral points on Markoff type cubic surfaces and dynamics Sarnak, Peter | CIRM H Nouveau

Heuristics for boundedness of ranks of elliptic curves Poonen, Bjorn | CIRM H Nouveau

Formulas for the limiting distribution of traces of Frobenius Lachaud, Gilles | CIRM H Nouveau

Women in numbers 2: research directions in number theory.BIRKS workshop WIN2Banff # november 6-11, 2011 David, Chantal ; Lalin, Matilde ; Manes, Michelle | American Mathematical Society;Centre De Recherches Mathematiques 2013 Nouveau

Trends in number theory.Fifth Spanish meeting on number theorySevilla # July 8-12, 2013 Chamizo, Fernando ; Guardia, Jordi ; Rojas-Leon, Antonio ; Tornero, José Maria | American Mathematical Society;Real Sociedad Matematica Espanola 2015 Nouveau

Seminar on Fermat :1993-1994 fields institute for research in mathematical sciences Murty, Vijaya Kumar | American Mathematical Society 1995 Nouveau

Séminaire Bourbaki. Vol. 2011/2012:exposés 1043-1058 | Société Mathématique de France 2013 Nouveau

Séminaire Bourbaki. Vol. 2002/2003 :exposés 909-923 | Société Mathématique de France 2003 Nouveau

Séminaire Bourbaki. Vol. 1993/94 :exposés 775-789 | Société Mathématique de France 1995 Nouveau

Séminaire Bourbaki. Vol. 1989-90exposés 715-729 | Société Mathématique de France 1990 Nouveau

O-minimality and diophantine geometry.Including lecture notes from the LMS-EPSRC short courseManchester # July 2013 Jones, Gareth O. ; Wilkie, A. J. | Cambridge University Press 2015 Nouveau

Number theory related to modular curves : Momose memorial volume.Barcelona-Boston-Tokyo number theory seminar, in memory of Fumiyuki MomoseBarcelona # May 21-23, 2012 Lario, Joan-Carles ; Murty, V. Kumar | American Mathematical Society 2018 Nouveau

Number theory :tradition and modernization#Feb. 12-16 Zhang, Wenpeng ; Tanigawa, Yoshio | Springer 2006 Nouveau

Number theoretic methods :future trends#proceedings of the second China-Japan seminar held Fukuoka#March 12-16 Kanemitsu, Shigeru ; Jia, Chaohua | Kluwer Academic Publishers 2002 Nouveau

Feynman amplitudes, periods and motives.International research workshop on periods and motives - a modern perspective on renormalizationMadrid # July 2-6, 2012 Alvarez-Consul, Luis ; Burgos Gil, José Ignacio ; Ebrahimi-Fard, Kurusch | American Mathematical Society 2015 Nouveau

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Women in numbers 2: research directions in number theory.
BIRKS workshop WIN2
Banff # november 6-11, 2011 David, Chantal ; Lalin, Matilde ; Manes, Michelle | American Mathematical Society;Centre De Recherches Mathematiques 2013 Nouveau

Trends in number theory.
Fifth Spanish meeting on number theory
Sevilla # July 8-12, 2013 Chamizo, Fernando ; Guardia, Jordi ; Rojas-Leon, Antonio ; Tornero, José Maria | American Mathematical Society;Real Sociedad Matematica Espanola 2015 Nouveau

Seminar on Fermat :
1993-1994 fields institute for research in mathematical sciences Murty, Vijaya Kumar | American Mathematical Society 1995 Nouveau

Séminaire Bourbaki. Vol. 2011/2012:
exposés 1043-1058 | Société Mathématique de France 2013 Nouveau

Séminaire Bourbaki. Vol. 2002/2003 :
exposés 909-923 | Société Mathématique de France 2003 Nouveau

Séminaire Bourbaki. Vol. 1993/94 :
exposés 775-789 | Société Mathématique de France 1995 Nouveau

Séminaire Bourbaki. Vol. 1989-90
exposés 715-729 | Société Mathématique de France 1990 Nouveau

O-minimality and diophantine geometry.
Including lecture notes from the LMS-EPSRC short course
Manchester # July 2013 Jones, Gareth O. ; Wilkie, A. J. | Cambridge University Press 2015 Nouveau

Number theory related to modular curves : Momose memorial volume.
Barcelona-Boston-Tokyo number theory seminar, in memory of Fumiyuki Momose
Barcelona # May 21-23, 2012 Lario, Joan-Carles ; Murty, V. Kumar | American Mathematical Society 2018 Nouveau

Number theory :
tradition and modernization#Feb. 12-16 Zhang, Wenpeng ; Tanigawa, Yoshio | Springer 2006 Nouveau

Number theoretic methods :
future trends#proceedings of the second China-Japan seminar held Fukuoka#March 12-16 Kanemitsu, Shigeru ; Jia, Chaohua | Kluwer Academic Publishers 2002 Nouveau

Feynman amplitudes, periods and motives.
International research workshop on periods and motives - a modern perspective on renormalization
Madrid # July 2-6, 2012 Alvarez-Consul, Luis ; Burgos Gil, José Ignacio ; Ebrahimi-Fard, Kurusch | American Mathematical Society 2015 Nouveau