The circle method and pairs of quadratic forms
We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.
The density of integral points on hypersurfaces of degree at least four
The density of rational points on a pfaff curve
This paper is concerned with the density of rational points on the graph of a non-algebraic pfaffian function.
The density of S-integral points in projective space with respect to a quadric
The equation n₁n₂ = n₃n₄, the gcd-sum function and the mean values of certain character sums
The Frobenius number of geometric sequences.
The intersection of a curve with algebraic subgroups in a product of elliptic curves
We consider an irreducible curve in , where is an elliptic curve and and are both defined over . Assuming that is not contained in any translate of a proper algebraic subgroup of , we show that the points of the union , where ranges over all proper algebraic subgroups of , form a set of bounded canonical height. Furthermore, if has Complex Multiplication then the set , for ranging over all algebraic subgroups of of codimension at least , is finite. If has no Complex Multiplication...
The method of Thue-Siegel for binary quartic forms
The number of solutions of linear equations in roots of unity
The resolution of the diophantine equation x(x+d)...(x+(k-1)d) = by² for fixed d
The simultaneous Pell equations y²-Dz² = 1 and x²-2Dz² = 1
Théorème d'irréductibilité de Hilbert effectif
Trivial points on towers of curves
In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.