Counting rational points over function fields

In this thesis, we are dealing with both arithmetic and geometric problems coming from the study of rational points with a particular focus on function fields over finite fields: (1) Using the circle method we produce upper bounds for the number of rational points of bounded height on diagonal cubic surfaces and fourfolds over Fq(t). This is based on joint work with Leonhard Hochfilzer. (2) We study rational points on smooth complete intersections X defined by cubic and quadratic hypersurfaces over Fq(t). We refine the Farey dissection of the “unit square” developed by Vishe [202] and use the circle method with a Kloosterman refinement to establish an asymptotic formula for the number of rational points of bounded height on X when dim(X) ≥ 23. Under the same hypotheses, we also verify weak approximation. (3) In joint work with Hochfilzer, we obtain upper bounds for the number of rational points of bounded height on del Pezzo surfaces of low degree over any global field. Our approach is to take hyperplane sections, which reduces the problem to uniform estimates for the number of rational points on curves. (4) We develop a version of the circle method capable of counting Fq-points on jet schemes of moduli spaces of rational curves on hypersurfaces. Combining this with a spreading out argument and a result of Mustaţă [150], this allows us to show that these moduli spaces only have canonical singularities under suitable assumptions on the degree and the dimension. In addition, we give an overview of guiding questions and conjectures in the field of rational points and explain the basic mechanism underlying the circle method.