Fermat’s last theorem over some small real quadratic fields

Freitas, Nuno; Siksek, Samir

doi:10.2140/ant.2015.9.875

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Fermat's last theorem over some small real quadratic fields

Nuno Freitas and Samir Siksek

Vol. 9 (2015), No. 4, 875–895

DOI: 10.2140/ant.2015.9.875

Abstract

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations, and ray class groups, we show that for $3 \leq d \leq 23$ , where $d \neq 5, 17$ and is squarefree, the Fermat equation $x^{n} + y^{n} = z^{n}$ has no nontrivial solutions over the quadratic field $ℚ (\sqrt{d})$ for $n \geq 4$ . Furthermore, we show that for $d = 17$ , the same holds for prime exponents $n \equiv 3, 5 (mod 8)$ .

Keywords

Fermat, modularity, Galois representation, level lowering

Mathematical Subject Classification 2010

Primary: 11D41

Secondary: 11F80, 11F03

Milestones

Received: 16 July 2014

Revised: 23 September 2014

Accepted: 9 March 2015

Published: 30 May 2015

Authors

Nuno Freitas
Mathematisches Institut Universität Bayreuth 95440 Bayreuth Germany

Samir Siksek
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom

Vol. 9, No. 4, 2015