## Algebra & Number Theory

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

#### 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

$∏ q∈S NormFq∕F5(u mod q)≠1̄.$

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 $n≠29,87,89$, then every semistable elliptic curve $E$ over $K$ is modular.

Let $ℓ,m,p$ be prime, with $ℓ,m ≥ 5$ and $p ≥ 3$. To a putative nontrivial primitive solution of the generalized Fermat equation $x2ℓ + y2m = zp$ we associate a Frey elliptic curve defined over $ℚ(ζp)+$, 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 $p ≤ 11$, or if $p = 13$ and $ℓ,m≠7$.

#### Article information

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

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

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

#### 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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