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The analytic rank of a family of jacobians of Fermat curves

Jędrzejak T.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 3-4

199-206

We study the family of curves Fm(p) : xp + yp = m, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves Fm(p). As a corollary we conclude that the Jacobians of the curves Fm(5) with even analytic rank and those with odd analytic rank are equally distributed.