## STNB2017(31è any)

### Asymptotic Fermat's Last Theorem over Number Fields

Haluk Sengun

#### Resum

Assuming two deep but standard reciprocity conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields $\mathbb{Q}(\sqrt{-d})$ with $-d \equiv 2, 3 \pmod{4}$. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem. This is joint work with Samir Siksek.

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