Isogenies, rational points and section points of group varieties

Rational points on varieties

Rational Points of Rationally Simply Connected Varieties

These are notes prepared for a series of lectures at the conference Variétés rationnellement connexes: aspects géométriques et arithmétiques of the Société Mathématique de France held in Strasbourg, France in May 2008.

Varieties with Too Many Rational Points

We investigate Fano varieties defined over a number field, which contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety.

Counting Rational Points on Algebraic Varieties

In these lectures we will be interested in solutions to Diophantine equations F (x1, . . . , xn) = 0, where F is an absolutely irreducible polynomial with integer coefficients, and the solutions are to satisfy (x1, . . . , xn) ∈ Z. Such an equation represents a hypersurface in A, and we may prefer to talk of integer points on this hypersurface, rather than solutions to the corresponding Diophan...

Counting Rational Points on Ruled Varieties

In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety V which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting fu...