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For a prime number $q\geq 5$ and a positive integer $N$ prime to $q$, Ribetproved the action of the Hecke algebra on the component group of the Jacobianvariety of the modular curve of level $Nq$ at $q$ is "Eisenstein", which meansthe Hecke operator $T_\ell$ acts by $\ell+1$ when $\ell$ is a prime number notdividing the level. In this paper, we completely compute the action of theHecke algebra on this component group by a careful study of supersingularpoints with extra automorphisms.

The action of the Hecke operators on the component groups of modular Jacobian varieties