Rational points on cubic hypersurfaces over $\mathbb{F}_q(t)$

TL;DR

This paper proves the Hasse principle and weak approximation for non-singular cubic hypersurfaces over the function field of a finite field, given the hypersurface's dimension is at least 6.

Contribution

It establishes the Hasse principle and weak approximation for cubic hypersurfaces over $ ext{finite field } k(t)$ with dimension ≥ 6, extending known results to this setting.

Findings

Hasse principle holds for cubic hypersurfaces over $k(t)$ with dimension ≥ 6.

Weak approximation is valid for these hypersurfaces.

Results depend on the characteristic of the finite field exceeding 3.

Abstract

For any finite field k of characteristic exceeding 3, the Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field k(t), provided that X has dimension at least 6.