STNB2019 (33è any)

On the asymptotic Fermat Last Theorem

Nuno Freitas

Resum

Let $K$ be a number field. The asymptotic Fermat's Last Theorem (AFLT) states that there exists a constant $B_K$, depending only on $K$, such that for all prime numbers $p > B_K$ all the solutions to the Fermat equation $x^p + y^p = z^p$ satisfy $xyz = 0$.

In this talk we will discuss how this relates to the non-existence of certain elliptic curves over $K$ and sketch the proof of AFLT over infinitely many fields, including all the layers of the $\mathbb{Z}_2$-extesion of $\mathbb{Q}$.

This is joint work with Alain Kraus and Samir Siksek.

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