The Analytic Rank of a Family of Jacobians of Fermat Curves
Volume 56 / 2008
Bulletin Polish Acad. Sci. Math. 56 (2008), 199-206
MSC: 11G40, 11G10, 11G30, 11G35.
DOI: 10.4064/ba56-3-2
Abstract
We study the family of curves $F_{m}( p) :x^{p}+y^{p}=m$, where $p$ is an odd prime and $m$ is a $p$th 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 $F_{m}(p) $. As a corollary we conclude that the jacobians of the curves $F_{m}( 5) $ with even analytic rank and those with odd analytic rank are equally distributed.
