Integral points on affine surfaces fibered over $\mathbb{A}^{1}$

TL;DR

This paper applies Swinnerton-Dyer's descent-fibration method, building on Harpaz's work, to analyze integral points on affine surfaces that are fibered by norm 1 tori over the affine line.

Contribution

It introduces a novel application of the descent-fibration method to a new class of affine surfaces, extending previous techniques to norm 1 torus fibrations.

Findings

Identification of conditions for the existence of integral points

Extension of descent-fibration methods to new surface classes

Results on the distribution of integral points on these surfaces

Abstract

Profitant du travail de pr\'ec\'edent d'Harpaz nous utilisons la m\'ethode de descente-fibration de Swinnerton-Dyer pour \'etudier les points int\'egraux sur des surfaces affines qui sont des fibration de tores de norme 1 sur $\mathbb{A}^{1}$. Taking advantage of previous work of Harpaz we use Swinnerton-Dyer's descent-fibration method to study integral points on affine surfaces which are fibrations of norm 1 tori over $\mathbb{A}^{1}$.