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

Taekyung Kim, Hwajong Yoo

Published 2017-10-17Version 1

For a prime number $q\geq 5$ 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 $T_\ell$ acts by $\ell+1$ when $\ell$ 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.