On the conjectures of Mordell and Lang in positive characteristics.
On the Difference of the Weil Height and the Néron-Tate Height.
On the distribution of rational points on certain Kummer surfaces
On the equation
On the equation a³ + b³ⁿ = c²
We study coprime integer solutions to the equation a³ + b³ⁿ = c² using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.
On the group of components of a Néron model.
On the group orders of elliptic curves over finite fields
On the Hasse principle for homogeneous spaces with finite stabilizers
On the method of Coleman and Chabauty.
On the Poincaré series associated to the p-adic points on a curve
On the Rational Points of Abelian Varieties over Zp Extensions of Number Fields.
On the Real Points of an Arithmetic Quotient of a Bounded Symmetrie Domain.
On the reducibility type of trinomials
On the solutions of certain diagonal quadratic equations and Lang's conjecture
On the torsion subgroups of elliptic curves over totally real fields.
On the value set of small families of polynomials over a finite field, II
We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in of degree d for which s consecutive coefficients are fixed. Our estimate asserts that , where . We also prove that , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of of degree d with s consecutive coefficients fixed as above. Finally, we show that , where ₂(d,0) denotes the average second moment for all monic polynomials...
Orbifolds, special varieties and classification theory
This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler...
Orbifolds, special varieties and classification theory: an appendix
For any compact Kähler manifold and for any equivalence relation generated by a symmetric binary relation with compact analytic graph in , the existence of a meromorphic quotient is known from Inv. Math. 63 (1981). We give here a simplified and detailed proof of the existence of such quotients, following the approach of that paper. These quotients are used in one of the two constructions of the core of given in the previous paper of this fascicule, as well as in many other questions.