Explicit descent for Jacobians of cyclic coevers of the projective line.
Explicit determination of the images of the Galois representations attached to abelian surfaces with .
Explicit global function fields over the binary field with many rational places
Explicit local heights.
Explicit moduli for curves of genus 2 with real multiplication by ℚ(√5)
1. Motivation. Let J₀(N) denote the Jacobian of the modular curve X₀(N) parametrizing pairs of N-isogenous elliptic curves. The simple factors of J₀(N) have real multiplication, that is to say that the endomorphism ring of a simple factor A contains an order in a totally real number field of degree dim A. We shall sometimes abbreviate "real multiplication" to "RM" and say that A has maximal RM by the totally real field F if A has an action of the full ring of integers of F. We say that a...
Explicit Selmer groups for cyclic covers of ℙ¹
For any abelian variety J over a global field k and an isogeny ϕ: J → J, the Selmer group is a subgroup of the Galois cohomology group , defined in terms of local data. When J is the Jacobian of a cyclic cover of ℙ¹ of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the ‘fake Selmer group’, whose definition is more amenable to explicit computations. In this paper we define in the same setting the ‘explicit Selmer group’, which is isomorphic...
Extensions diedrales et courbes elliptiques
Facteurs -simples de de grande dimension et de grand rang
Factoring polynomials in finite fields: an application of Lang-Weil to a problem in graph theory.
Factorization estimates for abelian varieties
Failure of the Hasse principle for Châtelet surfaces in characteristic
Given any global field of characteristic , we construct a Châtelet surface over that fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic , thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.
Fake CM and the stable model of .
Families of elliptic curves of high rank with nontrivial torsion group over ℚ
Families of hypersurfaces of large degree
Grauert and Manin showed that a non-isotrivial family of compact complex hyperbolic curves has finitely many sections. We consider a generic moving enough family of high enough degree hypersurfaces in a complex projective space. We show the existence of a strict closed subset of its total space that contains the image of all its sections.
Familles de polynômes liées aux courbes modulaires X₁(l) unicursales et points rationnels non-triviaux de courbes elliptiques quotient
Familles de revêtements de la droite projective
Fibrations of low genus, I
Field of moduli versus field of definition for cyclic covers of the projective line
We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli that can not be defined over is also given.
Fields of definition of building blocks with quaternionic multiplication
Fields of definition of -curves
Let be a -curve with no complex multiplication. In this note we characterize the number fields such that there is a curve isogenous to having all the isogenies between its Galois conjugates defined over , and also the curves isogenous to defined over a number field such that the abelian variety Res obtained by restriction of scalars is a product of abelian varieties of GL-type.