Une remarque sur les courbes de Reichardt-Lind et de Schinzel

TL;DR

This paper proves that the arithmetic fundamental group of the Schinzel curve has no section over the absolute Galois group of Q, providing evidence for Grothendieck's section conjecture in this specific case.

Contribution

It confirms the section conjecture for the Schinzel curve by showing the non-existence of a section of its arithmetic fundamental group over the absolute Galois group of Q.

Findings

No section exists for the fundamental group over the Galois group of Q.

Supports Grothendieck's section conjecture in this example.

Provides a specific case confirming theoretical predictions.

Abstract

We prove that the arithmetic fundamental group of X admits no section over the absolute Galois group of Q when X is the Schinzel curve, thereby confirming in this example the prediction given by Grothendieck's section conjecture. ----- Nous d\'emontrons que le groupe fondamental arithm\'etique de X n'admet pas de section au-dessus du groupe de Galois absolu de Q lorsque X est la courbe de Schinzel, confirmant ainsi sur cet exemple la pr\'ediction donn\'ee par la conjecture des sections de Grothendieck.