Vol. 4, No. 4, 2010

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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
Abstract

We determine the stable models of the modular curves X0(p3) for primes p 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/