The canonical height of an algebraic point on an elliptic curve.
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 equation x+y = α in multiplicatively dependent unknowns
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 Lehmer strength bounds for total ramification
The Mahler measure of dihedral extensions
The optimality of the Bounded Height Conjecture
In this article we show that the Bounded Height Conjecture is optimal in the sense that, if is an irreducible subvariety with empty deprived set in a power of an elliptic curve, then every open subset of does not have bounded height. The Bounded Height Conjecture is known to hold. We also present some examples and remarks.
Theta height and Faltings height
Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized Abelian variety. We also give as an application an explicit upper bound on the number of -rational points of a curve of genus under a conjecture of S. Lang and J. Silverman. We complete the study with a comparison between differential lattice structures.
Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)
In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples...