On the group orders of elliptic curves over finite fields
Everett W. Howe (1993)
Compositio Mathematica
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Everett W. Howe (1993)
Compositio Mathematica
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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...
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...
Noam D. Elkies (1990)
Compositio Mathematica
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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 of rational points of a curve of genus over a finite field , for a number of pairs . Given a pair and an integer , we determine the possible zeta functions of genus- curves over with points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus- curve over with points must have a low-degree map to another curve over , and often this...
J. E. Cremona (1987)
Compositio Mathematica
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Noam D. Elkies (1989)
Compositio Mathematica
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