Points of bounded height on quintic del Pezzo surfaces over number fields

TL;DR

This paper proves Manin's conjecture for split smooth quintic del Pezzo surfaces over any number field by analyzing height functions via universal torsors and o-minimal structures.

Contribution

It establishes the conjecture for a new class of surfaces using novel torsor size restrictions and o-minimal counting techniques.

Findings

Manin's conjecture verified for quintic del Pezzo surfaces

Effective height counting via universal torsors

Application of o-minimal structures to Diophantine geometry

Abstract

We prove Manin's conjecture for split smooth quintic del Pezzo surfaces over arbitrary number fields with respect to fairly general anticanonical height functions. After passing to universal torsors, we first show that we may restrict the torsor variables to their typical sizes, and then we can solve the counting problem in the framework of o-minimal structures.