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

By Gilles Lachaud, Christophe Ritzenthaler, Michael A. Tsfasman

This quantity comprises the court cases of the eleventh convention on AGC2T, held in Marseilles, France in November 2007. There are 12 unique study articles overlaying asymptotic homes of world fields, mathematics houses of curves and better dimensional types, and functions to codes and cryptography. This quantity additionally incorporates a survey article on purposes of finite fields by means of J.-P. Serre. AGC2T meetings happen in Marseilles, France each 2 years. those overseas meetings were an important occasion within the sector of utilized mathematics geometry for greater than twenty years.

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If there is no line of X ∩Q through the vertex of the cone Q = Π0 E3 , each line of Q intersecting X in at most two points, we deduce that |X ∩ Q| ≤ 2(q 2 + 1). e. 3, we have w(X , Q) = 4, 5. 1. Let us consider now that w(X , Q) = 4. If r(X )=3,4 and g(X )=2, or r(X )=4 and g(X )=1, from [5, IV-D-2, IV-E2] we get |X ∩ Q ∩ E3 (Fq )| ≤ 2(q + 1). 2 that |X ∩ Q| ≤ 2q 2 + 2q + 1. 4. e. Q = P4 ) and r(X )=3,4. 2. with Q at the place of X . (i). 2. Intersection of two non-degenerate quadrics. Here we study the number of points in the intersection of two non-degenerate quadrics X and Q.

If ζn ∈ / K, then the construction of Zn,−1 is more complicated; in this case ZN,−1 is still a quotient of X(n) × X(n), (where now X(n)/K denotes Shimura’s canonical model of X(n)/C), but the quotient has to be taken with respect to an ´etale group scheme rather than a (constant) group (scheme); cf. [Ka7] for more detail. 42 10 GERHARD FREY AND ERNST KANI For us it is important that we have found a very explicit variety which is isomorphic to the Hurwitz space Hn∗ (as we shall see), and this allows us to study its geometric and Diophantine properties.

Indeed the two secant lines (D1 ) and (D2 ) intersect in a point P . The plane P =< D1 , D2 > defined by these two secant lines through P only lies in the tangent hyperplane to X in P . Let H1 be this tangent hyperplane, Xˆ1 is a cone quadric and Xˆ1 ∩ Qˆ1 is of type 1 or 2. Therefore we get |Xˆ1 ∩ Qˆ1 | ≤ 4q + 1. 4) where (D1 ∪ D2 ) taking the place of D, we deduce that |X ∩ Q| ≤ 2q 2 + 3q + 1. 4. The structure of the code C2 (X ) defined on the quadric X |X | When X = Z(F ) ⊂ Pn (Fq ) is a quadric, the map c : F2 −→ Fq is not injective and we have: dim C2 (X ) = n(n+3) .

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