STNB

STNB2026 (39a edició)

Fermat curves over number fields and the Hasse principle

Ponents

Alain Kraus

Resum

Let $p$ be a prime number. A Fermat curve over $\mathbb{Q}$ of exponent $p$ is defined by an equation of the shape $ax^p+by^p+cz^p=0$ where $a,b,c$ are non-zero rational numbers. The aim of this talk is to discuss the following recent result. Let $d \geq 1$ be an integer. If $p$ is large enough with respect to $d$, for instance $p>1+d(d+1)$, then there exist infinitely many Fermat curves over $\mathbb{Q}$ of exponent p such that, for any number field $K$ of degree $[K : \mathbb{Q}] \leq d$, these curves contradict the Hasse principle over K and are pairwise non K-isomorphic.

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