The action of the hecke operators on the component groups of modular jacobian varieties

Abstract

For a prime number q≥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 Tℓ acts by ℓ C 1 when ℓ is a prime number not dividing the level. We completely compute the action of the Hecke algebra on this component group by a careful study of supersingular points with extra automorphisms. © 2018 Mathematical Sciences Publishers