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Abstract

Let $p$ be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over $\mathbb Q$, given by equations $ax^p+by^p+cz^p=0$, with respect to the local-global Hasse principle. It is conjectured that there exist infinitely many Fermat curves of exponent $p$ which are counterexamples to the Hasse principle. This is a consequence of the abc-conjecture if $p\geq 5$. Using a cyclotomic approach due to H. Cohen and Chebotarev’s density theorem, we obtain a partial result towards this conjecture, by proving it for $p\leq 19$.