Geodesic
$J_1(p)$ has connected fibers
Documenta mathematica, Tome 8 (2003), pp. 331-408

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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⊆(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 of arithmetic invariants of optimal quotients of J1​(p).
DOI : 10.4171/dm/146
Classification : 11F11, 11Y40, 14H40
Mots-clés : resolution of singularities, Jacobians of modular curves, component groups 
William Stein; Brian Conrad; Bas Edixhoven. $J_1(p)$ has connected fibers. Documenta mathematica, Tome 8 (2003), pp. 331-408. doi: 10.4171/dm/146
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     title = {$J_1(p)$ has connected fibers},
     journal = {Documenta mathematica},
     pages = {331--408},
     year = {2003},
     volume = {8},
     doi = {10.4171/dm/146},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/146/}
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