Displaying similar documents to “Reed-Muller codes and supersingular curves. I”

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...

On equations defining fake elliptic curves

Pilar Bayer, Jordi Guàrdia (2005)

Journal de Théorie des Nombres de Bordeaux

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Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as . We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As in the...

Improved upper bounds for the number of points on curves over finite fields

Everett W. Howe, Kristin E. Lauter (2003)

Annales de l’institut Fourier

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We give new arguments that improve the known upper bounds on the maximal number N q ( g ) of rational points of a curve of genus g over a finite field 𝔽 q , for a number of pairs ( q , g ) . Given a pair ( q , g ) and an integer N , we determine the possible zeta functions of genus- g curves over 𝔽 q with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus- g curve over 𝔽 q with N points must have a low-degree map to another curve over 𝔽 q , and often this...