The Asymptotic Fermat's Last Theorem for Five-Sixths of Real Quadratic Fields

TL;DR

This paper proves that for most real quadratic fields, there exists a threshold beyond which the Fermat equation has no nontrivial solutions, using modularity and number theory techniques.

Contribution

It introduces an algorithmic criterion for the asymptotic Fermat's Last Theorem over real quadratic fields and shows it holds for 5/6 of such fields, improving to 1 under conjecture.

Findings

Criterion satisfied by 5/6 of real quadratic fields

Potential to extend to all real quadratic fields under conjecture

Uses modularity and analytic number theory techniques

Abstract

Let $K$ be a totally real field. By the asymptotic Fermat's Last Theorem over $K$ we mean the statement that there is a constant $B_K$ such that for prime exponents $p>B_K$ the only solutions to the Fermat equation $a^p + b^p + c^p = 0$ with $a$, $b$, $c$ in $K$ are the trivial ones satisfying $abc = 0$. With the help of modularity, level lowering and image of inertia comparisons we give an algorithmically testable criterion which if satisfied by $K$ implies the asymptotic Fermat's Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $K = \mathbb{Q}(\sqrt{d})$ for a subset of $d$ having density $5/6$ among the squarefree positive integers. We can improve this to density 1 if we assume a standard "Eichler-Shimura" conjecture.