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J1(p) Has Connected Fibers

By Brian Conrad, Bas Edixhoven and William Stein


We study resolution of tame cyclic quotient singularities on arithmetic surfaces, and use it to prove that for any subgroup H ⊆ (Z/pZ) × /{±1} the map XH(p) = X1(p)/H → X0(p) induces an injection Φ(JH(p)) → Φ(J0(p)) on mod p component groups, with image equal to that of H in Φ(J0(p)) when the latter is viewed as a quotient of the cyclic group (Z/pZ) × /{±1}. In particular, Φ(JH(p)) is always Eisenstein in the sense of Mazur and Ribet, and Φ(J1(p)) is trivial: that is, J1(p) has connected fibers. We also compute tables o

Topics: Jacobians of modular curves, Component groups, Resolution of singularities Contents
Year: 2003
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