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

TL;DR

This paper derives formulas for root numbers of Jacobian varieties of Fermat curves, studies their distribution, bounds Selmer groups, and verifies the $p$-parity conjecture under certain conditions.

Contribution

It provides explicit root number formulas, analyzes their distribution, bounds Selmer groups, and confirms the $p$-parity conjecture for specific Fermat Jacobians.

Findings

Root numbers are explicitly computed for Fermat Jacobians.

Root numbers are shown to be equidistributed in families.

Selmer groups are bounded and explicitly determined under certain conditions.

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.