DOCUMENTA MATHEMATICA, Vol. 8 (2003), 331-408

Brian Conrad, Bas Edixhoven, William Stein

$J_1(p)$ Has Connected Fibers

We study resolution of tame cyclic quotient singularities on arithmetic surfaces, and use it to prove that for any subgroup $H \subseteq (\Z/p\Z)^{\times}/\{\pm 1\}$ the map $X_H(p) = X_1(p)/H \rightarrow X_0(p)$ induces an injection $\Phi(J_H(p)) \rightarrow \Phi(J_0(p))$ 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 $(\Z/p\Z)^{\times}/\{\pm 1\}$. 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 $J_1(p)$.

2000 Mathematics Subject Classification: 11F11, 11Y40, 14H40

Keywords and Phrases: Jacobians of modular curves, Component groups, Resolution of singularities

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