Vol. 10, No. 6, 2016

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ISSN: 1944-7833 (e-only)
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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
Abstract

Let K be a real abelian field of odd class number in which 5 is unramified. Let S5 be the set of places of K above 5. Suppose for every nonempty proper subset S S5 there is a totally positive unit u OK such that

qS NormFqF5(u mod q)1̄.

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 n29,87,89, then every semistable elliptic curve E over K is modular.

Let ,m,p be prime, with ,m 5 and p 3. To a putative nontrivial primitive solution of the generalized Fermat equation x2 + y2m = zp we associate a Frey elliptic curve defined over (ζp)+, and study its mod representation with the help of level lowering and our modularity result. We deduce the nonexistence of nontrivial primitive solutions if p 11, or if p = 13 and ,m7.

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