Rational and integral points of bounded height on surfaces over global fields

Manin’s conjecture predicts the distribution of rational points of bounded height on Fano varieties over number fields. Two methods to verify Manin’s conjecture for certain varieties are harmonic analysis on height zeta functions and the universal torsor method. We apply these two methods to obtain results for counting problems related to Manin’s conjecture: counting integral points on singular del Pezzo surfaces over number fields and rational points in families of conics over global function fields. Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch and formulated as a conjecture over log Fano varieties by Santens. Previously, it was proved for toric varieties and similar classes with a suitable group action, where harmonic analysis can be applied. For non-toric del Pezzo surfaces it was proved only over ℚ, using universal torsors. In the first part of the thesis, we generalise the torsor method without exploiting a group action for integral points from ℚ to arbitrary number fields. As representative examples, we count integral points of bounded loganticanonical height on a quartic del Pezzo surface of singularity type 𝐀₃ over imaginary quadratic fields and a quartic del Pezzo surface of singularity type 𝐀₃ + 𝐀₁ over arbitrary number fields, both with respect to its singularities and lines, using universal torsors. We interpret the count over number fields with one archimedean place in the framework of Chambert-Loir and Tschinkel geometrically to prove an analogue of Manin’s conjecture for integral points. Peyre proposed an analogue of Manin’s conjecture over global function fields. Serre was the first to consider conics in a family with rational points. Later, Loughran and Smeets generalised this work to certain families of varieties. They stated a conjecture about the asymptotic behaviour of the number of varieties over number fields with rational points in families. We investigate whether their conjecture also holds over global function fields: in the second part of the thesis, we count the number of conics in a family that have a rational point over the global function field 𝐾 = 𝔽₂(𝑡). By using harmonic analysis, we compute the height zeta function. We further prove a Tauberian Theorem for Dirichlet series with branch points, which we apply to obtain an asymptotic formula for the number of conics of bounded height in the considered family.