#### Vol. 296, No. 2, 2018

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

### Taekyung Kim and Hwajong Yoo

Vol. 296 (2018), No. 2, 341–355
##### Abstract

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 ${T}_{\ell }$ acts by $\ell +1$ when $\ell$ 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.

##### Keywords
Hecke operators, Hecke action, component group, modular Jacobian varieties
##### Mathematical Subject Classification 2010
Primary: 11G05, 11G18, 14G35