Problèmes de Skolem sur les champs algébriques

We extend Rumely's local-global principle (as refined by Cantor, Roquette, and the author, i.e. with local splitting conditions) to the case of algebraic stacks, in Artin's sense, over rings of (S-)integers of global fields. The nongeometrically connected case is also taken into account, as well as (in some instances) the case where local conditions are imposed at all places.     

Construction de surfaces algébriques dont le diviseur de Severi est supérieur à l'unité

Sans résuméAu Prof.van der Waerden, pour son 60^e anniversaire     

Erreurs de chute dans la résolution de systèmes algébriques linéaires

Sans résumé Institut de mathématiques appliquées de l’Ecole Polytechnique de l’Université de Lausanne. Recherche subventionnée par le Fonds national suisse de la recherche scientifique.     

Courbes algébriques et équations multiplicatives

We study the intersection of an algebraic curve C lying in a multiplicative torus over with the union of all algebraic subgroups of codimension 2. Finiteness of this set has already been proved by Bombieri, Masser and Zannier under the assumption that C is not contained in a translate of a proper subtorus. Following this result, the question of the minimal hypothesis implying finiteness has been raised by these authors, giving rise to the conjecture~: finiteness holds precisely when C is not contained in a proper subgroup. We prove here this statement which is also a special case of more general conjectures stated independently by Zilber and Pink. Our proof takes its inspiration from an article by Rémond and Viada concerning the Zilber-Pink conjecture for curves lying in a power of an elliptic curve. Hence, it relies on a uniform version of the Vojta inequality proven via the generalized Vojta inequality of Rémond. The main task is to establish a lower bound for some intersection numbers, here on a whole family of surfaces obtained by blowing up a compactification of C ×  C.     

Automorphismes des extensions algébriques d'un corps

Sans résumé     

Premiers chiffres significatifs et nombres algébriques

I show that there is a meromorphic continuation of σ_{logβn/a<logβa}n^−s to the whole plane if and only if β is a Pisot number, α ε Q(β), and either the second largest conjugate of β is real or the conjugate of α corresponding to the second largest conjugate of β is positive.     

Construction et classification de certaines solutions algébriques des systèmes de Garnier

IsomonodromicdeformationsofFuchsianequationsoforder2onRiemann sphere are parameterized by the solutions of Garnier system. The purpose of this paper is to construct algebraic solutions exotic, i.e. corresponding to deformations of Fuchsian equation with Zariski dense monodromy. Specifically, we classify all the algebraic solutions (complete) exotic constructed by the method of pull-back of Doran-Kitaev: they are deduced from the data isomonodromic deformations pulling back a Fuchsian equation E given by a family of branched coverings ? _t . We first introduce the structures and associated orbifoldes underlying Fuchsian equation. This allows us to have are fined version of the Riemann Hurwitz formula that allows us quickly to show that E must be hypergeometric. Then we come to limit the degree of ? and exponents, and finally to Painlevé VI. We explicitly construct one of these solutions.     

Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques

In this paper we study formal moduli for wildly ramified Galois covering. We prove a local-global principle. We then focus on the infinitesimal deformations of the Z/p Z-covers. We explicitly compute a deformation of an automorphism of order p which implies a universal obstruction for p>2. By deforming Artin-Schreier equations we obtain a lower bound on the dimension of the local versal deformation ring. At last, by comparing the global versal deformation ring to the complete local ring in a point of a moduli space, we determine the dimensions of the global and local versal deformation rings.

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Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques

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Oblatum 22-II-1999 & 15-XII-1999?Published online: 29 March 2000     

Sur un théorème de Kronecker concernant les variétés algébriques

A classical result of Kronecker, stated at the end of the Section 10 of Kronecker (J. Reine Angew. Math. 92 (1882) 1–123), is that any radical of a finitely generated ideal in a polynomial ring of n variables is the radical of an ideal generated by n+1 elements. We give a constructive and elementary proof of a generalisation presented in (Michigan Math. J. 31 (1984) (2) 167–180): in a ring of Krull dimension $\text{?}\text{n}$ a radical of a finitely generated ideal is the radical of an ideal generated by n+1 elements. To cite this article: T. Coquand, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Paramètres algébriques de groupes formels et critères d'isogénie

We generalise G. and D. Chudnovsky's isogeny criterion for elliptic curves over Q to the case of Abelian varieties of dimension g. The proof is bused on an explicit construction of algebraic parameters of their formal groups.     

Équirépartition adélique de mesures algébriques dans un groupe résoluble et sommes de Kloosterman

This paper answers a question of Clozel and Ullmo, showing that certain sequences of adelically-defined probability measures defined on an adelic quotient of a solvable group converge to the uniform measure on that quotient. This turns out to depend on any non-trivial estimate for classical Kloosterman sums. At the end, a “horizontal” analogue of the problem is stated and solved using a result of Duke, Friedlander and Iwaniec. Received: 30 November 2005     

Approximations simultanées de nombres algébriques de Q p par des rationnels

 In this paper, we prove that if β₁,…, β _n are p-adic numbers belonging to an algebraic number field K of degree n + 1 over Q such that 1, β₁,…,β _n are linearly independent over Z, there exist infinitely many sets of integers (q ₀,…, q _n ), with q ₀ ≠ 0 and

Sur les comitants algèbriques d'un objet de la connexion linéaire tensorielle

Zusammenfassung Nach Gh. Vranceanu und seinen Mitarbeitern kann man zu jeder in einem Koordinatensystem konstanten linearen übertragung eine hyperkomplexe Algebra zuordnen. Man kann aber auch umgekehrt von einer bestimmten hyperkomplexen Algebra ausgehen und die Differentialgeometrie der zugeh?rigen linearen übertragung untersuchen. Dafür wird hier der Schiefk?rper der Hamiltonschen Quaternionen gew?hlt. Enrico Bompiani zu seinem wissenschaftlichen Jubil?um.     

Formules pour les espaces multisécants aux courbes algébriques

We give a general new algorithm to find formulas for the number of k-secant d-planes to a smooth curve in ${\mathbf{P}}^N(\mathbf{C})$, for any k, d, N. Partial results were given by Castelnuovo [Rend. Palermo 3 (1889) 27–37] and Tanturri [Ann. Mat. 4 (1900) 67–122]. See also E. Arbarello et al. [Geometry of Algebraic Curves, vol. 1, Grundlehren Math. Wiss., vol. 267, Springer-Verlag, 1985]. To cite this article: P. Le Barz, C. R. Acad. Sci. Paris, Ser. I 340 (2005).