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

DOI: 10.2140/pjm.2018.296.341

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

Milestones

Received: 23 October 2017

Revised: 5 February 2018

Accepted: 6 February 2018

Published: 16 July 2018