Let p be a prime greater than 3 and not equal to 7, and let K be the cyclotomicfield obtained by adjoining a primitive p-th root of 1 to the rational numbers. We compute the image of the K-rational torsion part of the Mordell-Weil group of the Jacobian of the Fermat curve of exponent p under the standard isogeny to the product of the Jacobians of the cyclic Fermat quotients. We then use this to show that for certain (infinitely many) number fields M, there are no M-rational points in the cuspidal torsion packet of the Fermat curve other than the points at infinity.
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