Formelle Standardmodelle hyperelliptischer Kurven.
Formes automorphes et séries de Dirichlet
Formes compagnons et complexe BGG dual pour
On montre, sous certaines hypothèses un résultat en direction de la conjecture de Serre pour formulée dans un autre article avec F. Herzig : si la représentation résiduelle associée à une forme de Siegel de genre , de niveau premier à , -ordinaire de poids -petit, laisse stables deux droites (au lieu d’une) dans un plan lagrangien, alors cette forme possède une forme compagnon de poids prescrit. Notre méthode consiste à traduire, grâce au théorème de comparaison mod. de Faltings, l’existence...
Formes Différentielles et Modules de Tate des Variétés Abéliennes sur les Corps Locaux.
Formes linéaires de logarithmes effectives sur les variétés abéliennes
Formule de Lefschetz et rationalité des fonctions
Formules explicites et minorations de conducteurs de variétés algébriques
Formules explicites pour le nombre de points des variétés sur un corps fini.
Free pencils on divisors.
Frobenius indices of certain curves over finite fields.
From the elliptic regulator to exotic relations.
From the Taniyama-Shimura conjecture to Fermat's last theorem
Function Fields of Quadratic Forms.
Fundamental domains for Shimura curves
We describe a process for defining and computing a fundamental domain in the upper half plane of a Shimura curve associated with an order in a quaternion algebra . A fundamental domain for realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves . The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due to Michon....
Fundamental groups and Diophantine geometry
This is a brief exposition on the uses of non-commutative fundamental groups in the study of Diophantine problems.
Galois actions on Néron models of Jacobians
Let be a smooth curve defined over the fraction field of a complete discrete valuation ring . We study a natural filtration of the special fiber of the Néron model of the Jacobian of by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for over , and in particular are independent of the residue characteristic. Furthermore, we obtain information about...
Galois Coverings of the Non-Archimedean Projective Line.
Galois Covers and the Hilbert-Grunwald Property
Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a -adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over . The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.
Galois descent and cohomology for algebraic groups.
Galois orbits and equidistribution: Manin-Mumford and André-Oort.
We overview a unified approach to the André-Oort and Manin-Mumford conjectures based on a combination of Galois-theoretic and ergodic techniques. This paper is based on recent work of Klingler, Ullmo and Yafaev on the André-Oort conjecture, and of Ratazzi and Ullmo on the Manin-Mumford conjecture.