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 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 acts by + 1 when 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.

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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
Authors
Taekyung Kim
Center for Geometry and Physics
Institute for Basic Science
Pohang
South Korea
Hwajong Yoo
Center for Geometry and Physics
Institute for Basic Science
Pohang
South Korea