Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves

Let k be a field of characteristic different from 2. There can be an obstruction for an indecomposable principally polarized abelian threefold (A,a) over k to be a Jacobian over k. It can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numeric results to prove or refute the existence of some optimal curves of genus 3.Comment: 24 pages ; added : an explicit model, remarks on the hyperelliptic and decomposable reduction, reference

Complete addition laws on abelian varieties

We prove that under any projective embedding of an abelian variety A of dimension g, a complete system of addition laws has cardinality at least g+1, generalizing of a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in P^2. In contrast with this geometric constraint, we moreover prove that if k is any field with infinite absolute Galois group, then there exists, for every abelian variety A/k, a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or embedding in P^15, respectively, up to a finite number of counterexamples for |k| less or equal to 5.Comment: 9 pages. Finale version, accepted for publication in LMS Journal of Computation and Mathematic

Faster computation of the Tate pairing

This paper proposes new explicit formulas for the doubling and addition step in Miller's algorithm to compute the Tate pairing. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in the addition and doubling. Computing the coefficients of the functions and the sum or double of the points is faster than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also speed up pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves.Comment: 15 pages, 2 figures. Final version accepted for publication in Journal of Number Theor

Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic model over a degree 2 extension of this field.Comment: 35 pages; improve the readability of the pape

Methode A.G.M. pour les courbes ordinaires de genre 3 non hyperelliptiques sur F_{2^N}

We propose a A.G.M. algorithm for the determination of the characteristic polynomial of an ordinary non hyperelliptic curve of genus 3 over F_{2^N}.Comment: 8 pages, frenc

Optimal curves of genus 1,2 and 3

In this survey, we discuss the problem of the maximum number of points of curves of genus 1,2 and 3 over finite fieldsComment: 18 pages. To appear in "Publications Mathematiques de Besancon(PMB)