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arXiv · 1908.11690

On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one

George Catalin Turcas
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Abstract

In the present article, we extend previous results of the author and we show that when $K$ is any quadratic imaginary field of class number one, Fermat's equation $a^p+b^p+c^p=0$ does not have integral coprime solutions $a,b,c \in K \setminus \{ 0 \}$ such that $2 \mid abc$ and $p \geq 19$ is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.

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BibTeXRIS

George Catalin Turcas. 2019-08-28. On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one. https://doi.org/10.1016/j.jnt.2019.08.011

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