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The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields

Part of: Discontinuous groups and automorphic forms Diophantine equations

Published online by Cambridge University Press:  06 March 2015

Nuno Freitas  and
Samir Siksek
Nuno Freitas
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany email [email protected]
Samir Siksek
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email [email protected]
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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 any prime exponent $p>B_{K}$, the only solutions to the Fermat equation

$$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$
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 $d\geqslant 2$ having density ${\textstyle \frac{5}{6}}$ among the squarefree positive integers. We can improve this density to $1$ if we assume a standard ‘Eichler–Shimura’ conjecture.

Keywords

FermatmodularityGalois representationlevel lowering

MSC classification

Primary: 11D41: Higher degree equations; Fermat's equation
Secondary: 11F80: Galois representations 11F03: Modular and automorphic functions
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 8 , August 2015 , pp. 1395 - 1415
Copyright
© The Authors 2015 

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