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

TL;DR

This paper computes the action of Hecke operators on the component groups of modular Jacobian varieties at a prime, extending Ribet's Eisenstein property to a complete description using supersingular points.

Contribution

It provides a complete computation of the Hecke algebra action on component groups of modular Jacobians at prime levels, using supersingular points with automorphisms.

Findings

Hecke operators act in a predictable Eisenstein manner at prime levels.

The action is explicitly computed using properties of supersingular points.

The results extend previous partial understandings of the Hecke action.

Abstract

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.