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Algebra & Number Theory Algebra & Number Theory Volume 4 2010 No. 4 mathematical sciences publishers Algebra & Number Theory www.jant.org EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Andrei Okounkov Princeton University, USA John H. Coates University of Cambridge, UK Raman Parimala Emory University, USA J-L. Colliot-Thel´ ene` CNRS, Universite´ Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hel´ ene` Esnault Universitat¨ Duisburg-Essen, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universitat,¨ Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Ronald Solomon Ohio State University, USA Andrew Granville Universite´ de Montreal,´ Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Kansas, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University, USA Yujiro Kawamata University of Tokyo, Japan Michel van den Bergh Hasselt University, Belgium Janos´ Kollar´ Princeton University, USA Marie-France Vigneras´ Universite´ Paris VII, France Hendrik W. Lenstra Universiteit Leiden, The Netherlands Kei-Ichi Watanabe Nihon University, Japan Yuri Manin Northwestern University, USA Andrei Zelevinsky Northeastern University, USA Barry Mazur Harvard University, USA Efim Zelmanov University of California, San Diego, USA PRODUCTION [email protected] Silvio Levy, Scientific Editor Andrew Levy, Production Editor See inside back cover or www.jant.org for submission instructions. The subscription price for 2010 is US $140/year for the electronic version, and $200/year (C$30 shipping outside the US) for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA. Algebra & Number Theory (ISSN 1937-0652) at Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. ANT peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers http://www.mathscipub.org A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2010 by Mathematical Sciences Publishers ALGEBRA AND NUMBER THEORY 4:4(2010) 3 Stable reduction of X0(p ) Ken McMurdy and Robert Coleman with an appendix by Everett W. Howe This paper is dedicated to Siegfried Bosch, whose foundational work in rigid analysis was invaluable in our development of the theory of semistable coverings. 3 We determine the stable models of the modular curves X0.p / 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. 1. Introduction Let n be an integer and p a prime. It is known that if n ≥ 3 and p ≥ 5, or if n ≥ 1 n and p ≥ 11, the modular curve X0.p / does not have a model with good reduction over the ring of integers of any complete subfield of Cp. By a model for a scheme C over a complete local field K , we mean a scheme S over the ring of integers ᏻK ∼ ⊗ of K such that C D S ᏻK K . When a curve C over K does not have a model with good reduction over ᏻK , it may have the “next best thing”, that is, a stable model. The stable model is unique up to isomorphism if it exists, which it does over the ring of integers in some finite extension of K , as long as the genus of the curve is at least 2. Moreover, if Ꮿ is a stable model for C over ᏻK , and K ⊆ L ⊆ Cp, then ⊗ ⊗ Ꮿ ᏻK ᏻL is a stable model for C K L over ᏻL . The special fiber of any stable model for C is called the stable reduction. Here is a brief summary of prior results regarding the stable models of modular curves at prime power levels. Deligne and Rapoport [1973, §VI.6] found models for X0.p/ and X1.p/ over Zp and ZpTµpU that become stable over the quadratic unramified extension. Edixhoven [1990, Theorem 2.1.2] found stable models for 2 unr X0.p / over the ring of integers, R, in the Galois extension of Qp of degree .p2 − 1/=2. Bouw and Wewers [2004, Theorem 4.1 and Corollary 3.4] found stable models of X0.p/ and X .p/ over Zp and R by completely different means. n Krir [1996, Theor´ eme` 1] proved that the Jacobian of X0.p / has a semistable MSC2000: primary 14G22; secondary 11G07, 14G35. Keywords: stable reduction, modular curves, rigid analysis. 357 358 Ken McMurdy and Robert Coleman unr model over the ring of integers of an explicit Galois extension Ln of Qp of degree 2.n−2/ 2 n p .p − 1/ for n ≥ 2, which implies that X0.p / has a stable model over the ring of integers of Ln by [Deligne and Mumford 1969, Theorem 2.4]. Also, stable models for X0.125/ and X0.81/ were computed explicitly in [McMurdy 2004, §2; 4 2008, §3], and [2008, §5] gave a conjectural stable reduction of X0.p /. The main 3 result of this paper is the construction of a stable model for X0.p /, when p ≥ 13, over the ring of integers of some finite extension of Qp that is made explicit in [CM 2006]. We introduce the notion of a semistable covering of a smooth complete curve over a complete nonarchimedean field in Section 2C. We prove that any curve over a complete stable subfield of Cp has a semistable covering if and only if it has a semistable model, and moreover we can determine the corresponding reduction from the covering (see Theorem 2.36). Finding a semistable covering is often easier in practice than finding a semistable model directly, and this is what we do 3 for X0.p / in Sections 6–9. Overview. Our approach is rigid analytic, in that we construct a stable model of 3 X0.p / by actually constructing a stable covering by wide opens (an equivalent rigid analytic notion which was introduced in [Coleman and McCallum 1988, §1]). A covering Ꮿo of the ordinary locus can be obtained by extending the ordinary ± ± affinoids Xa b defined in [Coleman 2005, §1] to wide open neighborhoods Wa b. The supersingular locus essentially breaks up into the union of finitely many de- formation spaces of height 2 formal groups with level structure [Lubin et al. 1964]. We use results from [Hopkins and Gross 1994] and [de Shalit 1994] to produce a covering Ꮿs of this region. Finally, we show that the genus of the covering Ꮿo [Ꮿs 3 is at least the genus of X0.p /, and therefore that the overall covering is stable. This argument is laid out as follows. First, in Section 2, we recall or prove the general rigid analytic results that are necessary for a stable covering argument. These results are proved not only over complete subfields of Cp, but over more general complete nonarchimedean-valued fields. For example, Proposition 2.34 is the aforementioned result that the genus of any stable covering must equal the genus of the curve. We also revise and extend results of Bosch, and of Bosch and Lutkebohmert,¨ on the rigid geometry of algebraic curves. A rigid analytic version of the Riemann existence theorem is proved in Appendix A. In Section 3, we recall and fix notation for some results specifically pertaining n to X0.p / and its rigid subspaces. This is done from the moduli-theoretic point n of view, which is that points of X0.p / correspond to pairs .E; C/, where E is a generalized elliptic curve and C is a cyclic subgroup of order pn. There is a detailed discussion in Section 3A of the theory of the canonical subgroup of an elliptic curve Stable reduction of X0(p3) 359 and its connection with the geometry of X0.p/ [Buzzard 2003, §3]. Section 3B is ± ± where we define wide open neighborhoods, Wa b ⊇ Xa b, of the irreducible affinoids n that make up the ordinary locus of X0.p /. All of the necessary results regarding deformations of formal groups are given in Section 4. First we precisely state the relationship between deformations of elliptic curves and formal groups, which we call Woods Hole theory [Lubin et al. 1964, §6]. This is then used in Section 4A (along with the result of Howe in Appendix B) to prove that all of the connected components of the supersingular n locus of X0.p / are (nearly) isomorphic. Because of this fact, we are able to focus n on those regions WA.p / that correspond to a supersingular elliptic curve A=Fp for which j.A/ 6D 0; 1728. Specifically, this enables us to directly apply results of de Shalit [1994, §3] for the forgetful map from X0.p/ to the j-line. The other important consequence of Woods Hole theory is that it gives us a natural action of O ∼ ∗ Aut.A/ D .End.A/ ⊗ Zp/ n on WA.p /.
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