Un contre-exemple sur les families de courbes gauches

In this paper we give an example of two composants X and X,_o of the Hilbert scheme of space curves (i.e. components of subschemes with constant cohomology) satisfying the condition of semi-continuity (the cohomology of curves in X is less than the cohomology of curves in X,_o) but not the property X ₀?[Xbar]≠Ø.     

Degré des diviseurs sur les familles de courbes de ℙ3

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Fractions continues sur les surfaces de Veech

We define a geometrical continued fraction algorithm in the setting of regular polygons with an even number of sides., The definition of the algorithm uses linear transformations generating a group conjugated to an index 2 subgroup of a Hecke group. We give Markov conditions allowing the iteration of the algorithm. We compute the natural extension and the invariant measure for each of the additive and multiplicative versions of this algorithm.      

Points de petite hauteur sur les courbes modulaires X0(N)

Let N be a square free integer, prime to 6. Let φ the imbeding of X ₀(N) in its Jacobian relative to the point ∞. We show that the set is finite and that is infinite. This explicit form of the Bogomolov conjecture is obtained by an estimation of the self-intersection of the dualizing sheaf, in the sense of Arakelov theory, of modular curves. This result is obtained by estimating several quantities attached to the Arakelov metric on X ₀(N), starting with Petersson's trace formula Oblatum 1-I-1997 & 30-IV-1997     

Torsion sur des familles de courbes de genre g

We construct here, forl=2g ² +2g+1 or2g ² +3g+1, a family with one parameter of hyperelliptic curves of genusg overQ such that its jacobian has a point of orderl rational overQ(t). Wheng=2 the method allows to construct, forl=17, 19 or 21 a family with one parameter of hyperelliptic curves of genus 2 overQ such that its jacobian has a point of orderl rational overQ(t).      

Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète

Let be a hyperelliptic curve of genus over a discrete valuation field . In this article we study the models of over the ring of integers of . To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of over dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.

     

Approximation diophantienne sur les courbes elliptiques à multiplication complexe

Let $\text{E}$ be an elliptic curve with complex multiplication, defined over $\text{Q}$. We consider linear forms on $\text{Lie}\text{(}\text{E}{}^{\mathrm{n}}\text{)}$ with coefficients in the CM field of $\text{E}$. Within this framework, we present a new measure of linear independence for elliptic logarithms in (logb)(loga)ⁿ. Like recent advances in this domain (works by Ably, David, Hirata-Kohno), our result is best possible in terms of the height of the linear forms (logb) while providing a better estimate in the height of algebraic points considered (loga), removing a term in logloga. To cite this article: M. Ably, É. Gaudron, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,I

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,II

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur les modules de torsion des courbes elliptiques

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Re?u le 1 janvier 1997     

Descente infinie et hauteurp-adique sur les courbes elliptiques à multiplication complexe

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Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques

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Sur les revêtements de Schottky des courbes modulaires de Drinfeld

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