Fermat’s Last Theorem and modular curves over real quadratic fields
Volume 203 / 2022
Acta Arithmetica 203 (2022), 319-351
MSC: Primary 11D41; Secondary 11F80, 11G18, 11G05, 14G05.
DOI: 10.4064/aa210812-2-4
Published online: 9 May 2022
Abstract
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 $n \geq 4$.
