$J_1(p)$ has connected fibers

By Brian Conrad, Bas Edixhoven, William A. Stein. Documenta Mathematica, 2003

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

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e2b605a7c0b435001d91555fdb910cfd96cb4e21