Root numbers and Selmer groups for the Jacobian varieties of Fermat curves

Abstract

Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=\delta$ where $\delta$ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. When $p\nmid \delta$, we bound the Selmer groups of these Jacobian varieties. Moreover, if $p$ is regular and all prime ideals of $K$ dividing $\delta$ are inert in $K/F$, the Selmer groups are explicitly determined and we verify the $p$-parity conjectures of these Jacobian varieties. We also give an asymptotic lower bound for the number of Fermat Jabobians for which the $p$-parity conjecture holds.