Rational points of bounded height and the Weil restriction

TL;DR

This paper investigates the relationship between counting rational points of bounded height on a variety over a number field and its Weil restriction, proving new cases of conjectures and constructing counterexamples.

Contribution

It introduces methods to compare rational point counts under Weil restriction and proves new instances of Manin's conjecture, also providing counterexamples over various fields.

Findings

Proved several new cases of Manin's conjecture for Weil restrictions.

Constructed counterexamples over every number field.

Established compatibility of asymptotic formulas with Weil restriction.

Abstract

Given an extension of number fields $E \subset F$ and a projective variety $X$ over $F$, we compare the problem of counting the number of rational points of bounded height on $X$ with that of its Weil restriction over $E$. In particular, we consider the compatibility with respect to the Weil restriction of conjectural asymptotic formulae due to Manin and others. Using our methods we prove several new cases of these conjectures. We also construct new counterexamples over every number field.