Weak Approximation for Cubic Hypersurfaces and Degree 4 del Pezzo Surfaces

TL;DR

This paper proves new results on weak approximation for smooth cubic hypersurfaces and degree 4 del Pezzo surfaces over global fields, establishing conditions under which weak approximation holds at places of good reduction.

Contribution

It establishes weak approximation results for cubic hypersurfaces and del Pezzo surfaces over global function fields with specific residual field size conditions.

Findings

Weak approximation holds for cubic hypersurfaces over global function fields with residual field size ≥11.

Weak approximation holds for degree 4 del Pezzo surfaces with residual field size ≥13.

Weak approximation applies to high-dimensional cubic hypersurfaces over certain global fields.

Abstract

In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 11 elements. (2) For del Pezzo surfaces of degree 4 defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 13 elements. (3) Weak approximation holds for cubic hypersurfaces of dimension at least 10 defined over a global function field of characteristic not equal to 2, 3, 5 or a purely imaginary number field.