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

@article{Anni2015ModularEC, title={Modular elliptic curves over real abelian fields and the generalized Fermat equation \$x^\{2\ell\}+y^\{2m\}=z^p\$}, author={Samuele Anni and Samir Siksek}, journal={arXiv: Number Theory}, year={2015} }

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular.
Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a putative non-trivial primitive solution of the generalized Fermat…

## 9 Citations

### On some generalized Fermat equations of the form x2+y2n=zp$x^2+y^{2n} = z^p$

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The primary aim of this paper is to study the generalized Fermat equation x2+y2n=z3p\begin{equation*} x^2+y^{2n} = z^{3p} \end{equation*}in coprime integers x, y, and z, where n⩾2$n \geqslant 2$ and…

### Le théorème de Fermat sur certains corps de nombres totalement réels

- MathematicsAlgebra & Number Theory
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Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$,…

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- 2018

We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer $n \geq 2$, the equation \[ x^{13} + y^{13} = 3 z^n…

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Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent…

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- 2020

. In this paper, we consider ﬁve proofs related to the super-generalised Fermat equation, Pa x + Qb y = Rc z . All proofs depend on a new identity for a x + b y which can be expressed as a binomial…

### Ju l 2 02 1 On some Generalized Fermat Equations of the form x 2 + y 2 n = z p

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The primary aim of this paper is to study the generalized Fermat equation x + y = z in coprime integers x, y, and z, where n ≥ 2 and p is a fixed prime. Using modularity results over totally real…

### On elliptic curves with $p$-isogenies over quadratic fields

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Let E be an elliptic curve deﬁned over a number ﬁeld K . For which primes p does E admit a K -rational p -isogeny? Although we have an answer to this question over the rationals, extending this to…

### Proof of the Tijdeman-Zagier Conjecture via Slope Irrationality and Term Coprimality

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The Tijdeman-Zagier conjecture states no integer solution exists for AX + BY = CZ with positive integer bases and integer exponents greater than 2 unless gcd(A,B,C) > 1. Any set of values that…

### Points of order 13 on elliptic curves

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