$J_1(p)$ 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 (\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) .

Locations

• Documenta Mathematica - View - PDF