Fermat's Last Theorem and modular curves over real quadratic fields

TL;DR

This paper extends the study of Fermat's Last Theorem to quadratic fields with specific squarefree discriminants, using modular curves and Hilbert newforms to show the absence of non-trivial solutions for most cases when n ≥ 4.

Contribution

It introduces new methods involving quadratic points on modular curves and Hecke operators to prove non-existence of solutions over certain quadratic fields, extending prior work.

Findings

Most quadratic fields with 26 ≤ d ≤ 97 have no non-trivial solutions for n ≥ 4.

Utilizes analysis of quadratic points on modular curves and Hecke operators.

Extends previous results by Freitas and Siksek to a broader class of quadratic fields.

Abstract

In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.