$J_1(p)$ has connected fibers

Abstract

We study resolution of tame cyclic quotient singularities on arithmetic surfaces, and use it to prove that for any subgroup $H\subseteq (\mathbb{Z}/p\mathbb{Z}{)}^{\times }/\{\pm 1\}$ the map $X_H(p)=X_1(p)/H\to X_0(p)$ induces an injection $\Phi (J_H(p))\to \Phi (J_0(p))$ on mod $p$ component groups, with image equal to that of $H$ in $\Phi (J_0(p))$ when the latter is viewed as a quotient of the cyclic group $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }/\{\pm 1\}$. In particular, $\Phi (J_H(p))$ is always Eisenstein in the sense of Mazur and Ribet, and $\Phi (J_1(p))$ is trivial: that is, $J_1(p)$ has connected fibers. We also compute tables of arithmetic invariants of optimal quotients of $J_1(p)$.