STNB - Seminari de Teoria de Nombres de Barcelona - STNB2020(34è any) - A multi-Frey approach to Fermat equations of signature~$(7,7,p)$ combining Frey varieties of different dimensions.

STNB2020(34è any)

A multi-Frey approach to Fermat equations of signature~$(7,7,p)$ combining Frey varieties of different dimensions.

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Resum

We will discuss how various ideas from previous lectures combine to study Fermat type equations of signature $(7,7,p)$. More precisely, using a multi-Frey approach with two Frey elliptic curves over totally real fields, a Frey hyperelliptic over~$\Q$ due to Kraus and ideas from the Darmon program, we will give a complete resolution of the equation $$x^7 + y^7 = 3 z^n$$ for all integers $n \ge 2$.

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