On Elkies subgroups of $\ell $-torsion points in elliptic curves defined over a finite field

Reynald Lercier; Thomas Sirvent

Displaying similar documents to “On Elkies subgroups of $ℓ$ -torsion points in elliptic curves defined over a finite field”

$S$ -integral points on elliptic curves - Notes on a paper of B. M. M. de Weger

Emanuel Herrmann, Attila Pethö (2001)

Journal de théorie des nombres de Bordeaux

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In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and $p$ -adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

Counting points on elliptic curves over finite fields

René Schoof (1995)

Journal de théorie des nombres de Bordeaux

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We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large ; it is based on Shanks's baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic...

Counting points in medium characteristic using Kedlaya's algorithm.

Gaudry, Pierrick, Gürel, Nicolas (2003)

Experimental Mathematics

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On Manin's conditional algorithm

Horst G. Zimmer (1977)

Mémoires de la Société Mathématique de France

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No $17$ -torsion on elliptic curves over cubic number fields

Pierre Parent (2003)

Journal de théorie des nombres de Bordeaux

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We complete our previous determination of the torsion primes of elliptic curves over cubic number fields, by showing that $17$ is not one of those.

A new construction of $p$ -adic $L$ -functions attached to certain elliptic curves with complex multiplication

John L. Boxall (1986)

Annales de l'institut Fourier

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In this paper we apply the results of our previous article on the $p$ -adic interpolation of logarithmic derivatives of formal groups to the construction of $p$ -adic $L$ -functions attached to certain elliptic curves with complex multiplication. Our results are primarily concerned with curves with supersingular reduction.

Computing modular degrees using $L$ -functions

Christophe Delaunay (2003)

Journal de théorie des nombres de Bordeaux

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We give an algorithm to compute the modular degree of an elliptic curve defined over $ℚ$ . Our method is based on the computation of the special value at $s = 2$ of the symmetric square of the $L$ -function attached to the elliptic curve. This method is quite efficient and easy to implement.

A note on integral points on elliptic curves

Mark Watkins (2006)

Journal de Théorie des Nombres de Bordeaux

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We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of , with there being a unique rational nondegenerate solution. For the second...

A fast algorithm for polynomial factorization over $ℚ_{p}$

David Ford, Sebastian Pauli, Xavier-François Roblot (2002)

Journal de théorie des nombres de Bordeaux

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We present an algorithm that returns a proper factor of a polynomial $Φ (x)$ over the $p$ -adic integers $ℤ_{p}$ (if $Φ (x)$ is reducible over $ℚ_{p}$ ) or returns a power basis of the ring of integers of $ℚ_{p} [x] / Φ (x) ℚ_{p} [x]$ (if $Φ (x)$ is irreducible over $ℚ_{p}$ ). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.

Displaying similar documents to “On Elkies subgroups of ℓ -torsion points in elliptic curves defined over a finite field”

Displaying similar documents to “On Elkies subgroups of $ℓ$ -torsion points in elliptic curves defined over a finite field”