Twisted Fermat curves over totally real fields

TL;DR

This paper proves that for certain totally real fields, infinitely many twisted Fermat curves have no rational points over those fields, highlighting new obstructions to rational solutions in algebraic geometry.

Contribution

It establishes the existence of infinitely many twists of Fermat curves over specific totally real fields that lack rational points, extending understanding of rational solutions.

Findings

Infinitely many classes [delta] lead to curves with no F-rational points.

The result applies to fields with specific degree and Galois properties.

Provides new examples of obstructions to rational points on algebraic curves.

Abstract

Let p be a prime number, F a totally real field such that [F(mu_p): F]=2 and [F:Q] is odd. For delta \in F^times, let [delta] denote its class in F^times/F^{times p}. In this paper, we show Main Theorem. There are infinitely many classes [delta]\in F^times/F^{times p} such that the twisted affine Fermat curves W_delta: X^p+Y^p=delta have no F-rational points.