Arithmetic, Geometry, Cryptography and Coding Theory: by Gilles Lachaud, Christophe Ritzenthaler, Michael A. Tsfasman

By Gilles Lachaud, Christophe Ritzenthaler, Michael A. Tsfasman

This quantity includes the court cases of the eleventh convention on AGC2T, held in Marseilles, France in November 2007. There are 12 unique learn articles protecting asymptotic homes of worldwide fields, mathematics homes of curves and better dimensional kinds, and functions to codes and cryptography. This quantity additionally encompasses a survey article on functions of finite fields by means of J.-P. Serre. AGC2T meetings occur in Marseilles, France each 2 years. those foreign meetings were a tremendous occasion within the quarter of utilized mathematics geometry for greater than twenty years

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Additional resources for Arithmetic, Geometry, Cryptography and Coding Theory: International Conference November 5-9, 2007 Cirm, Marseilles, France

Example text

To explain this connection, note first that the points P1 , · · · , P4 determine an elliptic curve E/K which is unique up to a quadratic twist. This curve is constructed as follows. Choose a coordinate x of P1K such that (x)∞ = P4 , and a cubic polynomial f3 (x) ∈ K[x] with (f3 )0 = P1 + P2 + P3 . Then the (Weierstraß) equation y 2 = f3 (x) defines an elliptic curve E/K and a double cover π : E → P1 which is ramified at P1 , . . , P4 and which maps the zero-point 0E ∈ E(K) to P4 . Thus, the group of 2-torsion points of E is E[2] = {0E , P1 , P2 , P3 }, where Pi ∈ π −1 (Pi ) is the (unique) point above Pi .

If rank (X )=4 and g(X ) = 2, the result of [5, IV-B] gives |X ∩Q∩E3 (Fq )| = 4q or |X ∩ Q ∩ E3 (Fq )| ≤ 3q + 1. 2, we get either |X ∩ Q| = 4q 2 + 1 when each hyperplane is tangent to X with the plane of intersection meeting X at two lines or |X ∩ Q| ≤ 3q 2 + q + 1 otherwise. 3, we get r(X )=3. Here, the quadric Q describes a pair of lines in E2 (Fq ). The number of points in the intersection of two secant lines with a conic (non-singular plane quadric) is exactly four or less than three. 2 we get that either |X ∩ Q| = 4q 2 + q + 1 or |X ∩ Q| ≤ 3q 2 + q + 1.

Acknowledgment The author would like to thank Prof. F. Rodier and Prof. H. Van Maldeghem for their precious remarks and comments on a preliminary version of this paper. References [1] Y. Aubry, Reed-Muller codes associated to projective algebraic varieties. In “Algebraic Geomertry and Coding Theory ”. (Luminy, France, June 17-21, 1991). Lecture Notes in Math. 1518, Springer-Verlag, Berlin, (1992), 4-17. [2 ]J. Ax, Zeroes of polynomials over finite fields. Amer. J. , 86 (1964), 255-261. [3] M. Boguslavsky, On the number of solutions of polynomial systems.

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