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Algebra & Number Theory Vol. 7 (2013) Algebra & Number Theory Volume 7 2013 No. 3 msp Algebra & Number Theory msp.org/ant 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 Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Edward Frenkel University of California, Berkeley, USA Michael Singer North Carolina State University, USA Andrew Granville Université de Montréal, 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 Virginia, 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 János Kollár Princeton University, USA Marie-France Vignéras Université Paris VII, France Yuri Manin Northwestern University, USA Kei-Ichi Watanabe Nihon University, Japan Barry Mazur Harvard University, USA Efim Zelmanov University of California, San Diego, USA Philippe Michel École Polytechnique Fédérale de Lausanne PRODUCTION [email protected] Silvio Levy, Scientific Editor See inside back cover or msp.org/ant for submission instructions. The subscription price for 2013 is US $200/year for the electronic version, and $350/year (C$40, if shipping outside the US) for print and electronic. Subscriptions, requests for back issues and changes of subscribers address should be sent to MSP. Algebra & Number Theory (ISSN 1944-7833 electronic, 1937-0652 printed) at Mathematical Sciences Publishers, 798 Evans Hall #3840, c/o 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 nonprofit scientific publishing http://msp.org/ © 2013 Mathematical Sciences Publishers ALGEBRA AND NUMBER THEORY 7:3 (2013) msp dx.doi.org/10.2140/ant.2013.7.507 Ekedahl–Oort strata of hyperelliptic curves in characteristic 2 Arsen Elkin and Rachel Pries Suppose X is a hyperelliptic curve of genus g defined over an algebraically closed field k of characteristic p D 2. We prove that the de Rham cohomology of X decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the 2-torsion group scheme JX T2U of the Jacobian of X in terms of the Ekedahl–Oort type. The interesting feature is that JX T2U depends only on some discrete invariants of X, namely, on the ramification invariants associated with the branch points. We give a complete classification of the group schemes that occur as the 2-torsion group schemes of Jacobians of hyperelliptic k-curves of arbitrary genus, showing that only relatively few of the possible group schemes actually do occur. 1. Introduction Suppose k is an algebraically closed field of characteristic p > 0. There are several important stratifications of the moduli space Ꮽg of principally polarized abelian varieties of dimension g defined over k, including the Ekedahl–Oort stratification. The Ekedahl–Oort type characterizes the p-torsion group scheme of the correspond- ing abelian varieties and, in particular, determines invariants of the group scheme such as the p-rank and a-number. It is defined by the interaction between the Frobenius F and Verschiebung V operators on the p-torsion group scheme. Very little is known about how the Ekedahl–Oort strata intersect the Torelli locus of Jacobians of curves. In particular, one would like to know which group schemes occur as the p-torsion JX TpU of the Jacobian JX of a curve X of genus g. In this paper, we completely answer this question for hyperelliptic k-curves X of arbitrary genus when k has characteristic p D 2, a case that is amenable to calculation because of the confluence of hyperelliptic and Artin–Schreier properties. We first 1 prove a decomposition result about the structure of HdR.X/ as a module under the Elkin is partially supported by the Marie Curie Incoming International Fellowship PIIF-GA-2009- 236606. Pries is partially supported by National Science Foundation grant DMS-11-01712. MSC2010: primary 11G20; secondary 14K15, 14L15, 14H40, 14F40, 11G10. Keywords: curve, hyperelliptic, Artin–Schreier, Jacobian, p-torsion, a-number, group scheme, de Rham cohomology, Ekedahl–Oort strata. 507 508 Arsen Elkin and Rachel Pries actions of F and V , where the pieces of the decomposition are indexed by the branch points of the hyperelliptic cover. This is the only decomposition result about the de Rham cohomology of Artin–Schreier curves that we know of, though the action of V on H0.X; 1/ and the action of F on H1.X; ᏻ/ have been studied for Artin– Schreier curves under less restrictive hypotheses[Madden 1978; Sullivan 1975]. The second result of this paper is a complete classification of the isomorphism classes of group schemes that occur as the 2-torsion group scheme JX T2U for a hyperelliptic k-curve X of arbitrary genus when char.k/ D 2. The group schemes that occur decompose into pieces indexed by the branch points of the hyperelliptic cover, and we determine the Ekedahl–Oort types of these pieces. In particular, we determine which a-numbers occur for the 2-torsion group schemes of hyperelliptic k-curves of arbitrary genus when char.k/ D 2. Before describing the result precisely, we note that it shows that the group scheme JX T2U depends only on some discrete invariants of X and not on the location of the branch points or the equation of the hyperelliptic cover. This is in sharp contrast to the case of hyperelliptic curves in odd characteristic p, where even the p-rank depends on the location of the branch points[Yui 1978]. Notation 1.1. Suppose k is an algebraically closed field of characteristic p D 2. Let X be a k-curve of genus g that is hyperelliptic, in other words, for which there exists a degree two cover π V X ! P1. Let B ⊂ P1.k/ denote the set of branch points of π, and let r VD #B − 1. After a fractional linear transformation, one may suppose that 0 2 B and 1 2= B. For α 2 B, the ramification invariant dα is the largest integer for which the higher ramification group of π above α is nontrivial. By[Stichtenoth 2009, Propo- −1 sition III.7.8], dα is odd. Let cα VD .dα − 1/=2, and let xα VD .x − α/ . The cover π is given by an affine equation of the form y2 − y D f .x/ for some nonconstant rational function f .x/ 2 k.x/. After a change of variables of the form y 7! y C , one may suppose the partial fraction decomposition of f .x/ has the form X f .x/ D fα.xα/; (1-1) α2B 2 where fα.x/ 2 xkTx U is a polynomial of degree dα containing no monomials of even exponent. In particular, the divisor of poles of f .x/ on P1 has the form X div1. f .x// D dαα. α2B By the Riemann–Hurwitz formula[Serre 1968, IV, Proposition 4], the genus g of X satisfies X 2g C 2 D .dα C 1/: α2B Ekedahl–Oort strata of hyperelliptic curves in characteristic 2 509 T U Recall that the 2-rank of (the Jacobian of) the k-curve X is dimF2 Hom(µ2; JX 2 /, where µ2 is the kernel of Frobenius on Gm. By the Deuring–Shafarevich formula [Subrao 1975, Theorem 4.2; Crew 1984, Corollary 1.8], the 2-rank of X is r. Note P that g D r C α2B cα. The implication of these formulas is that, for a given genus g (and 2-rank r), there is an additional discrete invariant of the hyperelliptic k-curve X, namely, a partition of 2g C 2 into r C 1 positive even integers dα C 1. In Section 5a, we show that the Ekedahl–Oort type of X depends only on this discrete invariant. Theorem 1.2. Suppose X is a hyperelliptic curve defined over an algebraically closed field k of characteristic 2 with affine equation y2 − y D f .x/, branch locus B, and polynomials fα for α 2 B as described in Notation 1.1. For α 2 B, consider the 2 Artin–Schreier k-curve Yα with affine equation y −y D fα.x/. Let E be an ordinary elliptic k-curve. As a module under the actions of Frobenius F and Verschiebung V , the de Rham cohomology of X decomposes as 1 ∼ 1 #B−1 M 1 HdR.X/ D HdR.E/ ⊕ HdR.Yα/: α2B As an application of Theorem 1.2, we give a complete classification of the Ekedahl–Oort types that occur for hyperelliptic k-curves. Recall that the 2-torsion group scheme JX T2U of the Jacobian of a k-curve is a polarized BT1 group scheme over k (short for polarized Barsotti–Tate truncated level-1 group scheme) and that the isomorphism class of a BT1 group scheme determines and is determined by its Ekedahl–Oort type; see Section 2 for more details.
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