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# Documents  Cremona, John | enregistrements trouvés : 2

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

Post-edited

Number Theory

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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## Modular curves and abelian varieties :conference held at the Centre de Recerca Matematica Bellaterra#July 15-18 Cremona, John ; Lario, Joan-Carles ; Quer, Jordi ; Ribet, Kenneth A. | Birkhäuser 2004

Congrès

- 289 p.
ISBN 978-3-7643-6586-8

Progress in mathematics , 0224

Localisation : Collection 1er étage

courbe modulaire # forme modulaire # variété abélienne # conjecture de Shimura-Taniyama # courbe elliptique # représentation de Galois # courbe elliptique # variété abélienne de type GL2

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