By Gilles Lachaud, Christophe Ritzenthaler, Michael A. Tsfasman

ISBN-10: 0821847163

ISBN-13: 9780821847169

This quantity comprises the court cases of the eleventh convention on AGC2T, held in Marseilles, France in November 2007. There are 12 unique examine articles masking asymptotic houses of worldwide fields, mathematics homes of curves and better dimensional kinds, and purposes to codes and cryptography. This quantity additionally features a survey article on purposes of finite fields via J.-P. Serre. AGC2T meetings happen in Marseilles, France each 2 years. those foreign meetings were an enormous occasion within the zone of utilized mathematics geometry for greater than twenty years

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

**Example text**

P3 denote the images of the non-zero 2-torsion points of E under π and put P4 = π(0E ). In addition, let P5 ∈ P1 (K) be the unique point such that Disc(f ) = π ∗ (P5 ). e. if and only P5 ∈ / {P1 , . . , P4 }. Proof. Everything except the last statement has been explained above. e. not contained in {P1 , . . , P4 }. Note that the above construction guarantees that P1 , . . , P4 are distinct points, but not necessarily P1 , . . , P5 . 2). Then ϕ deﬁnes a point in the Hurwitz space Hn∗ . The previous propositions suggest that the moduli space Hn∗ is closely related to another moduli space Hn , which represents the moduli problem Hn classifying CURVES OF GENUS 2 WITH ELLIPTIC DIFFERENTIALS 41 9 isomorphism classes of pairs (C, f ) where C is a curve of genus 2 and f is a normalized covering map from C to an elliptic curve E of degree n.

From the result of D. B. Leep and L. M. 172] , we deduce the following result. 1. Let Q1 and Q2 be two quadrics in P4 (Fq ) with w(Q1 , Q2 ) = 5. Then, |Q1 ∩ Q2 | ≤ 3q 2 + q + 1. 172] the hypothesis on the order of the two quadrics Q1 and Q2 is too restrictive and does not work in the general case. Let us recall another property, free from this condition. 2. 70-71 ] Let Q1 and Q2 be two quadrics in Pn (Fq ) and l an integer such that 1 ≤ l ≤ n−1. e. , xn−l ) and |Q1 ∩ Q2 ∩ En−l (Fq )| ≤ m. Then |Q1 ∩ Q2 | ≤ mq l + πl−1 .

E. , xn−l ) and |Q1 ∩ Q2 ∩ En−l (Fq )| ≤ m. Then |Q1 ∩ Q2 | ≤ mq l + πl−1 . This bound is optimal as soon as m is optimal in En−l (Fq ). 3. 71 ] For Q1 and Q2 two quadrics in Pn (Fq ), we get w(Q1 , Q2 ) ≥ sup{r(Q1 ), r(Q2 )}. Let us recall the classiﬁcation of quadrics in P4 (Fq ) according to J. W. P. 4]. 132-136] we deduce that a non-degenerate quadric P4 in P4 (Fq ) contains exactly αq = π3 lines and there are exactly q + 1 lines contained in P4 through a given point in P4 . 4], is the set of transversals of three skew lines in P3 (Fq ); it consists of q +1 skew lines.

### Arithmetic, Geometry, Cryptography and Coding Theory: International Conference November 5-9, 2007 Cirm, Marseilles, France by Gilles Lachaud, Christophe Ritzenthaler, Michael A. Tsfasman

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