Algebra & Number Theory

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

Samuele Anni and Samir Siksek

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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.

Article information

Source
Algebra Number Theory, Volume 10, Number 6 (2016), 1147-1172.

Dates
Received: 9 June 2015
Revised: 22 March 2016
Accepted: 22 June 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842549

Digital Object Identifier
doi:10.2140/ant.2016.10.1147

Mathematical Reviews number (MathSciNet)
MR3544293

Zentralblatt MATH identifier
06626472

Subjects
Primary: 11D41: Higher degree equations; Fermat's equation 11F80: Galois representations
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]

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

Citation

Anni, Samuele; Siksek, Samir. Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$. Algebra Number Theory 10 (2016), no. 6, 1147--1172. doi:10.2140/ant.2016.10.1147. https://projecteuclid.org/euclid.ant/1510842549


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