Stable reduction of X0(p3)

McMurdy, Ken; Coleman, Robert

doi:10.2140/ant.2010.4.357

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Stable reduction of $X_0(p^3)$

Ken McMurdy and Robert Coleman

Appendix: Everett W. Howe

Vol. 4 (2010), No. 4, 357–431

DOI: 10.2140/ant.2010.4.357

Abstract

We determine the stable models of the modular curves $X_{0} (p^{3})$ for primes $p \geq 13$ . An essential ingredient is the close relationship between the deformation theories of elliptic curves and formal groups, which was established in the Woods Hole notes of 1964. This enables us to apply results of Hopkins and Gross in our analysis of the supersingular locus.

This paper is dedicated to Siegfried Bosch, whose foundational work in rigid analysis was invaluable in our development of the theory of semistable coverings.

Keywords

stable reduction, modular curves, rigid analysis

Mathematical Subject Classification 2000

Primary: 14G22

Secondary: 11G07, 14G35

Milestones

Received: 10 April 2007

Revised: 1 October 2009

Accepted: 9 October 2009

Published: 13 June 2010

Authors

Ken McMurdy
Department of Mathematics (TAS) Ramapo College of New Jersey 505 Ramapo Valley Rd. Mahwah, NJ 07430 United States
http://phobos.ramapo.edu/~kmcmurdy
Robert Coleman
Department of Mathematics University of California Berkeley, CA 94720 United States
http://math.berkeley.edu/~coleman/
Everett W. Howe
Center for Communications Research 4320 Westerra Court San Diego, CA 92121-1969 United States
http://alumnus.caltech.edu/~however/

Vol. 4, No. 4, 2010