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 $N\ge 1$ be an integer, and let $\hat{I}´\ge 1$ 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 $y^{{\ell }^N}=x^r(\hat{I}´-x{)}^s$ under the assumptions that $\ell \nmid rst$ and that ${\mathrm{ord}}_{\ell }(\hat{I}´)=0$ or $\ell \nmid {\mathrm{ord}}_{\ell }(\hat{I}´)$. 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 ${\ell }^{N-1}\Vert r$ and $\ell \nmid st\hat{I}´$.

9 pages