Fermat's Last Theorem and modular curves over real 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 $26\le d\le 97$. 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 $n\ge 4$.

Minor corrections for some computational errors

Cited by in corpus (6)

• Computing quadratic points on modular curves $X_0(N)$
• On elliptic curves with $p$-isogenies over quadratic fields
• $\mathbb{Q}$-curves and the Lebesgue-Nagell equation
• Computing points on bielliptic modular curves over fixed quadratic fields
• On some Generalized Fermat Equations of the form $x^2+y^{2n}=z^p$
• Fermat's Last Theorem over $\mathbb{Q}(\sqrt{2},\sqrt{3})$