The absolute anabelian geometry of canonical curves.
The distribution of group structures on elliptic curves over finite prime fields.
The fluctuations in the number of points on a family of curves over a finite field
Let be a positive integer, a finite field of cardinality with . In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of -points on the curve given by the affine model , where is drawn at random uniformly from the set of all monic -th power-free polynomials of degree as . The method also enables us to study the fluctuations in the number of -points on the same family of curves arising from the set of monic irreducible...
The genus of curves over finite fields with many rational points.
The hyperadelic gamma function.
The intrinsic Hodge theory of -adic hyperbolic curves.
The Mordell–Lang question for endomorphisms of semiabelian varieties
The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang...
The number of points on certain cubic surfaces over a finite field.
The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve
The Tate pairing for Abelian varieties over finite fields
In this expository note, we describe an arithmetic pairing associated to an isogeny between Abelian varieties over a finite field. We show that it generalises the Frey–Rück pairing, thereby giving a short proof of the perfectness of the latter.
The Weil pairing and the Hilbert symbol.
Traces of high powers of the Frobenius class in the hyperelliptic ensemble
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.