Root numbers for twisted Fermat quotient curves

Abstract

Let $\ell$ be an odd prime, $N\ge 1$ be an integer, and $\delta \ge 1$ be a ${\ell }^N$-th power free integer such that ${\mathrm{ord}}_{\ell }(\delta )=0$ or $\ell \nmid {\mathrm{ord}}_{\ell }(\delta )$. In this paper, we give an explicit formula for the root number of the Hecke character associated with a certain quotient curve of the twisted Fermat curve $X^{{\ell }^N}+Y^{{\ell }^N}=\delta$. This result gives a generalization of Stoll (2002) and Shu (2021).

arXiv comment31 pages. Conjectures for Fleck numbers (Conjecture 6.1 in v1) have been solved, and the main theorem has been simplified