Counting rational points on low-degree del Pezzo surfaces with automorphic forms

Abstract

We study Manin's conjecture for certain families of low-degree del Pezzo surfaces, i.e. asymptotics for the number of rational points of increasing height. In general, problems about rational points on del Pezzo surfaces are considered harder as the degree gets lower. In this thesis, we resolve Manin's conjecture for all Ch^atelet surfaces over Q, which are del Pezzo surfaces of degree four, and establish the first asymptotic count for a set of rational points on a del Pezzo surface of degree one. In Chapter 2, we prove the key analytic ingredient necessary in both arguments -- a bound on the sums of the absolute values of the Hecke eigenvalues of cuspidal automorphic forms of GL_2(A_Q) along polynomial values. We use pieces of the Sato-Tate distribution to prove a logarithmic power savings over the trivial bound in the applicable cases; additionally we classify, when the polynomial is solvable, exactly when a logarithmic power savings occurs using analysis about the base change of the cuspidal representation. In Chapter 3, we study Manin's conjecture for Ch^atelet surfaces -- the proper smooth models of the affine surfaces x^2+\Delta y^2 = f(z), where f(z) is a squarefree polynomial of degree 3 or 4. Previous works handle the case of \Delta=1 (and extend to any \Delta>0 such that Q(\sqrt{-\Delta}) has class number one). We resolve the conjecture by viewing the point-count through an automorphic lens and connecting it to the sums of Chapter 2. In Chapter 4, we study rational points on the singular del Pezzo surface of degree one defined by y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,where 4A^3-27B^2\neq 0 and Q(u,v) is a positive-definite quadratic form. The main algebraic tool used is a parameterization of the integral points on quadratic twists of elliptic curves by binary quartic forms; this parameterization was first observed by Mordell and uses syzygys of binary quartic forms. We then reduce the problem to correlation sums of binary quadratic and binary quartic forms, which are analyzed using the central analytic ingredients of Chapters 2 and 3. As a result, we establish an asymptotic count for rational points that are integral with respect to the singularity on these del Pezzo surfaces.