150 research outputs found
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)
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