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# Documents  Ballet, Stéphane | enregistrements trouvés : 9

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

Post-edited

Research talks

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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## Algorithmic arithmetic, geometry, and coding theory.14th International Conference on arithmetic, geometry, cryptography and coding theoryMarseille # June 3-7, 2013 Ballet, Stéphane ; Perret, Marc ; Zaytsev, Alexey | American Mathematical Society 2015

Congrès

- v; 306 p.
ISBN 978-1-4704-1461-0

Contemporary mathematics , 0637

Localisation : Collection 1er étage

théorie des codes # géométrie algébrique # cryptographie # théorie des nombres

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## Towers of Ramanujan graphs Li, Winnie | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry;Number Theory

A $d$-regular graph is Ramanujan if its nontrivial eigenvalues in absolute value are bounded by $2\sqrt{d-1}$. By means of number-theoretic methods,infinite families of Ramanujan graphs were constructed by Margulis and independently by Lubotzky-Phillips-Sarnak in 1980's for $d=q+ 1$, where q is a prime power. The existence of an infinite family of Ramanujan graphs for arbitrary d has been an open question since then. Recently Adam Marcus, Daniel Spielman and Nikhil Srivastava gave a positive answer to this question by showing that any bipartite $d$-regular Ramanujan graph has a $2$-fold cover that is also Ramanujan. In this talk we shall discuss their approach and mentionsimilarities with function field towers. A $d$-regular graph is Ramanujan if its nontrivial eigenvalues in absolute value are bounded by $2\sqrt{d-1}$. By means of number-theoretic methods,infinite families of Ramanujan graphs were constructed by Margulis and independently by Lubotzky-Phillips-Sarnak in 1980's for $d=q+ 1$, where q is a prime power. The existence of an infinite family of Ramanujan graphs for arbitrary d has been an open question since then. Recently Adam Marcus, Daniel ...

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## Good recursive towers Bassa, Alp | CIRM H

Multi angle

Research talks

Curves over finite fields of large genus with many rational points have been of interest for both theoretical reasons and for applications. In the past, various methods have been employed for the construction of such curves. One such method is by means of explicit recursive equations and will be the emphasis of this talk.The first explicit examples were found by Garcia-Stichtenoth over quadratic finite fields in 1995. Afterwards followed the discovery of good towers over cubic finite fields and finally all nonprime finite fields in 2013 (B.-Beelen-Garcia-Stichtenoth). The recursive nature of these towers makes them very special and in fact all good examples have been shown to have a modular interpretation of some sort. The questions of finding good recursive towers over prime fields resisted all attempts for several decades and lead to the common belief that such towers might not exist. In this talk I will try to give an overview of the landscape of explicit recursive towers and present a recently discovered tower over all finite fields including prime fields, except $F_{2}$ and $F_{3}$.
This is joint work with Christophe Ritzenthaler.
Curves over finite fields of large genus with many rational points have been of interest for both theoretical reasons and for applications. In the past, various methods have been employed for the construction of such curves. One such method is by means of explicit recursive equations and will be the emphasis of this talk.The first explicit examples were found by Garcia-Stichtenoth over quadratic finite fields in 1995. Afterwards followed the ...

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## The coset leader weight enumerator of the code of the twisted cubic Pellikaan, Ruud | CIRM H

Multi angle

Research talks

In general the computation of the weight enumerator of a code is hard and even harder so for the coset leader weight enumerator. Generalized Reed Solomon codes are MDS, so their weight enumerators are known and its formulas depend only on the length and the dimension of the code. The coset leader weight enumerator of an MDS code depends on the geometry of the associated projective system of points. We consider the coset leader weight enumerator of $F_{q}$-ary Generalized Reed Solomon codes of length q + 1 of small dimensions, so its associated projective system is a normal rational curve. For instance in case of the $\left [ q+1,3,q-1 \right ]_{q}$ code where the associated projective system of points consists of the q + 1 points of a plane conic, the answer depends whether the characteristic is odd or even. If the associated projective system of points of a $\left [ q+1,4,q-2 \right ]_{q}$ code consists of the q + 1 points of a twisted cubic, the answer depends on the value of the characteristic modulo 6. In general the computation of the weight enumerator of a code is hard and even harder so for the coset leader weight enumerator. Generalized Reed Solomon codes are MDS, so their weight enumerators are known and its formulas depend only on the length and the dimension of the code. The coset leader weight enumerator of an MDS code depends on the geometry of the associated projective system of points. We consider the coset leader weight enumerator ...

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## Ore polynomials and application to coding theory Caruso, Xavier | CIRM H

Multi angle

Research talks

In the 1930’s, in the course of developing non-commutative algebra, Ore introduced a twisted version of polynomials in which the scalars do not commute with the variable. About fifty years later, Delsarte, Roth and Gabidulin realized (independently) that Ore polynomials could be used to define codes—nowadays called Gabidulin codes—exhibiting good properties with respect to the rank distance. More recently, Gabidulin codes have received much attention because of many promising applications to network coding, distributed storage and cryptography.
The first part of my talk will be devoted to review the classical construction of Gabidulin codes and present a recent extension due to Martinez-Penas and Boucher (independently), offering similar performances but allowing for transmitting much longer messages in one shot. I will then revisit Martinez-Penas’ and Boucher’s constructions and give to them a geometric flavour. Based on this, I will derive a geometric description of duals of these codes and finally speculate on the existence of more general geometric Gabidulin codes.
In the 1930’s, in the course of developing non-commutative algebra, Ore introduced a twisted version of polynomials in which the scalars do not commute with the variable. About fifty years later, Delsarte, Roth and Gabidulin realized (independently) that Ore polynomials could be used to define codes—nowadays called Gabidulin codes—exhibiting good properties with respect to the rank distance. More recently, Gabidulin codes have received much ...

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## Gilles Lachaud: friend and mathematician Tsfasman, Michael | CIRM H

Multi angle

Research talks

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## Isolated points on modular curves Viray, Bianca | CIRM H

Multi angle

Research talks

Faltings’s theorem on rational points on subvarieties of abelian varieties can be used to show that al but finitely many algebraic points on a curve arise in families parametrized by $\mathbb{P}^{1}$ or positive rank abelian varieties, we call these finitely many exceptions isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on $X_{1}(n)$ push down to isolated points on aj only on the $j$-invariant of the isolated point.
This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.
Faltings’s theorem on rational points on subvarieties of abelian varieties can be used to show that al but finitely many algebraic points on a curve arise in families parametrized by $\mathbb{P}^{1}$ or positive rank abelian varieties, we call these finitely many exceptions isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on $X_{1}(n)$ push down ...

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## On the work and persona of Gilles Lachaud Ghorpade, Sudhir | CIRM H

Multi angle

Research talks

I will give an account of some aspects of the mathematical work of Gilles Lachaud, especially the work in which I was associated with him. This will be mixed with some personal reminiscences.

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