O-minimality on twisted universal torsors and Manin's conjecture over number fields

Abstract

Manin's conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several specific varieties over $\mathbb{Q}$, in particular del Pezzo surfaces. We show how this method can be implemented over arbitrary number fields $K$, by proving Manin's conjecture for a singular quartic del Pezzo surface of type $\mathbf{A}_3+\mathbf{A}_1$. The parameterization step is treated in high generality with the help of twisted integral models of universal torsors. To make the counting step feasible over arbitrary number fields, we deviate from the usual approach over $\mathbb{Q}$ by placing higher emphasis on the geometry of numbers in the framework of o-minimal structures.