Rational Points in Flag Varieties over Function Fields

Counting rational points on ruled varieties over function fields

Let K be the function field of an algebraic curve C defined over a finite field Fq. Let V ⊂ PK be a projective variety which is a union of lines. We prove a general result computing the number of rational points of bounded height on V/K. We first compute the number of rational points on a general line defined over K, and then sum over the lines covering V . Mathematics Subject Classification: 1...

Groups of Rational Points on Abelian Varieties over Finite Fields

Fix an isogeny class of abelian varieties with commutative endomorphism algebra over a finite field. This isogeny class is determined by a Weil polynomial fA without multiple roots. We give a classification of groups of rational points on varieties from this class in terms of Newton polygons of fA(1− t).

Remarks about Uniform Boundedness of Rational Points over Function Fields

We prove certain uniform versions of the Mordell Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points.

Rational Points on Curves over Finite Fields

Preface These notes treat the problem of counting the number of rational points on a curve defined over a finite field. The notes are an extended version of an earlier set of notes Aritmetisk Algebraisk Geometri – Kurver by Johan P. Hansen [Han] on the same subject. In Chapter 1 we summarize the basic notions of algebraic geometry, especially rational points and the Riemann-Roch theorem. For th...

Rational points on varieties