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

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $J_1(p)$.
Classification : 11F11, 11Y40, 14H40
Keywords: Jacobians of modular curves, component groups, resolution of singularities 
@article{DOCMA_2003__8__a7,
     author = {Conrad, Brian and Edixhoven, Bas and Stein, William},
     title = {$J_1(p)$ has connected fibers},
     journal = {Documenta mathematica},
     pages = {331--408},
     publisher = {mathdoc},
     volume = {8},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2003__8__a7/}
}
TY  - JOUR
AU  - Conrad, Brian
AU  - Edixhoven, Bas
AU  - Stein, William
TI  - $J_1(p)$ has connected fibers
JO  - Documenta mathematica
PY  - 2003
SP  - 331
EP  - 408
VL  - 8
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DOCMA_2003__8__a7/
LA  - en
ID  - DOCMA_2003__8__a7
ER  - 
%0 Journal Article
%A Conrad, Brian
%A Edixhoven, Bas
%A Stein, William
%T $J_1(p)$ has connected fibers
%J Documenta mathematica
%D 2003
%P 331-408
%V 8
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DOCMA_2003__8__a7/
%G en
%F DOCMA_2003__8__a7
Conrad, Brian; Edixhoven, Bas; Stein, William. $J_1(p)$ has connected fibers. Documenta mathematica, Tome 8 (2003), pp. 331-408. http://geodesic.mathdoc.fr/item/DOCMA_2003__8__a7/