Root numbers for twisted Fermat quotient curves II

Abstract

This is a sequel to the previous work of the author Yanagihara (2025). Let $\ell$ be an odd prime, let $N \geq 1$ be an integer, and let $δ\geq 1$ be an $\ell^N$-th-power-free integer. Let $r,s,t>0$ be integers satisfying $r+s+t=\ell^N$. In Yanagihara (2025), the author computed the root number of the Fermat quotient curve $y^{\ell^N}=x^r(δ-x)^s$ under the assumptions that $\ell\nmid rst$ and that $\operatorname{ord}_{\ell}(δ)=0$ or $\ell\nmid \operatorname{ord}_{\ell}(δ)$. In this paper, we study the case where the technical assumption $\ell\nmid rst$ is dropped. As one such case, we compute the root number when $\ell^{N-1}\| r$ and $\ell\nmid stδ$.