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

For a prime number $q\ge 5$ and a positive integer $N$ prime to $q$, Ribet proved the action of the Hecke algebra on the component group of the Jacobian variety of the modular curve of level $Nq$ at $q$ is “Eisenstein”, which means the Hecke operator $T_{\mathrm{\ell }}$ acts by $\mathrm{\ell }+1$ when $\mathrm{\ell }$ is a prime number not dividing the level. In this paper, we completely compute the action of the Hecke algebra on this component group by a careful study of supersingular points with extra automorphisms.