Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ + y2m = zp

Anni, Samuele; Siksek, Samir

doi:10.2140/ant.2016.10.1147

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Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$

Samuele Anni and Samir Siksek

Vol. 10 (2016), No. 6, 1147–1172

DOI: 10.2140/ant.2016.10.1147

Abstract

Let $K$ be a real abelian field of odd class number in which $5$ is unramified. Let $S_{5}$ be the set of places of $K$ above $5$ . Suppose for every nonempty proper subset $S \subset S_{5}$ there is a totally positive unit $u \in O_{K}$ such that

\prod_{q \in S} {Norm}_{F_{q} ∕ F_{5}} (u mod q) \neq \bar{1} .

We prove that every semistable elliptic curve over $K$ is modular, using a combination of several powerful modularity theorems and class field theory. We deduce that if $K$ is a real abelian field of conductor $n < 100$ , with $5 ∤ n$ and $n \neq 29, 87, 89$ , then every semistable elliptic curve $E$ over $K$ is modular.

Let $ℓ, m, p$ be prime, with $ℓ, m \geq 5$ and $p \geq 3$ . To a putative nontrivial primitive solution of the generalized Fermat equation $x^{2 ℓ} + y^{2 m} = z^{p}$ we associate a Frey elliptic curve defined over $ℚ {(ζ_{p})}^{+}$ , and study its mod $ℓ$ representation with the help of level lowering and our modularity result. We deduce the nonexistence of nontrivial primitive solutions if $p \leq 11$ , or if $p = 13$ and $ℓ, m \neq 7$ .

Keywords

elliptic curves, modularity, Galois representation, level lowering, irreducibility, generalized Fermat, Fermat–Catalan, Hilbert modular forms

Mathematical Subject Classification 2010

Primary: 11D41, 11F80

Secondary: 11G05, 11F41

Milestones

Received: 9 June 2015

Revised: 22 March 2016

Accepted: 22 June 2016

Published: 30 August 2016

Authors

Samuele Anni
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom

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

Vol. 10, No. 6, 2016