Elliptic 𝐾3 surfaces with 𝐩^{𝐧}-torsion sections

Ito, Hiroyuki; Liedtke, Christian

doi:10.1090/S1056-3911-2012-00584-0

Elliptic $\text {K3}$ surfaces with ${\mathbf p}^{\mathbf n}$-torsion sections

Authors: Hiroyuki Ito and Christian Liedtke
Journal: J. Algebraic Geom. 22 (2013), 105-139
DOI: https://doi.org/10.1090/S1056-3911-2012-00584-0
Published electronically: March 14, 2012
MathSciNet review: 2993049
Full-text PDF

Abstract | References | Additional Information

Abstract: We classify elliptic K3 surfaces in characteristic $p$ with $p^n$-torsion section. For $p^n\geq 3$ we verify conjectures of Artin and Shioda, compute the heights of their formal Brauer groups, as well as Artin invariants and Mordell–Weil groups in the supersingular cases.

References

M. Artin, Supersingular $K3$ surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975). MR 371899
M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 1, 87–131. MR 457458
Wolf Barth and Klaus Hulek, Projective models of Shioda modular surfaces, Manuscripta Math. 50 (1985), 73–132. MR 784140, DOI https://doi.org/10.1007/BF01168828
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
Richard M. Crew, Etale $p$-covers in characteristic $p$, Compositio Math. 52 (1984), no. 1, 31–45. MR 742696
I. Dolgachev and J. Keum, Wild $p$-cyclic actions on $K3$-surfaces, J. Algebraic Geom. 10 (2001), no. 1, 101–131. MR 1795552
Igor V. Dolgachev and JongHae Keum, Finite groups of symplectic automorphisms of $K3$ surfaces in positive characteristic, Ann. of Math. (2) 169 (2009), no. 1, 269–313. MR 2480606, DOI https://doi.org/10.4007/annals.2009.169.269
G. van der Geer and T. Katsura, On the height of Calabi-Yau varieties in positive characteristic, Doc. Math. 8 (2003), 97–113. MR 2029163
Luc Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501–661 (French). MR 565469
Jun-ichi Igusa, On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan 20 (1968), 96–106. MR 240103, DOI https://doi.org/10.2969/jmsj/02010096
Hiroyuki Ito, On extremal elliptic surfaces in characteristic 2 and 3, Hiroshima Math. J. 32 (2002), no. 2, 179–188. MR 1925896
Hiroyuki Ito, Deformation of a singularity of type $E_8$ and Mordell-Weil lattices in characteristic 2, Math. Nachr. 283 (2010), no. 7, 1037–1053. MR 2677286, DOI https://doi.org/10.1002/mana.200710057
Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569
William E. Lang, Extremal rational elliptic surfaces in characteristic $p$. II. Surfaces with three or fewer singular fibres, Ark. Mat. 32 (1994), no. 2, 423–448. MR 1318540, DOI https://doi.org/10.1007/BF02559579
Christian Liedtke and Stefan Schröer, The Néron model over the Igusa curves, J. Number Theory 130 (2010), no. 10, 2157–2197. MR 2660886, DOI https://doi.org/10.1016/j.jnt.2010.03.016
Rick Miranda and Ulf Persson, Torsion groups of elliptic surfaces, Compositio Math. 72 (1989), no. 3, 249–267. MR 1032336
Keiji Oguiso and Tetsuji Shioda, The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Paul. 40 (1991), no. 1, 83–99. MR 1104782
Arthur Ogus, Supersingular $K3$ crystals, Journées de Géométrie Algébrique de Rennes (Rennes, 1978) Astérisque, vol. 64, Soc. Math. France, Paris, 1979, pp. 3–86. MR 563467
Duc Tai Pho and Ichiro Shimada, Unirationality of certain supersingular $K3$ surfaces in characteristic 5, Manuscripta Math. 121 (2006), no. 4, 425–435. MR 2282430, DOI https://doi.org/10.1007/s00229-006-0045-3

A. N. Rudakov, I. R. Šafarevič, Supersingular K3 surfaces over fields of characteristic $2$, Math. USSR, Izv. 13, No. 1, 147-165 (1979).

Stefan Schröer, Kummer surfaces for the self-product of the cuspidal rational curve, J. Algebraic Geom. 16 (2007), no. 2, 305–346. MR 2274516, DOI https://doi.org/10.1090/S1056-3911-06-00438-3

Andreas Schweizer, On the $p^e$-torsion of elliptic curves and elliptic surfaces in characteristic $p$, Trans. Amer. Math. Soc. 357 (2005), no. 3, 1047–1059. MR 2110432, DOI https://doi.org/10.1090/S0002-9947-04-03520-2

Tetsuji Shioda, Kummer surfaces in characteristic $2$, Proc. Japan Acad. 50 (1974), 718–722. MR 491728

Tetsuji Shioda, An example of unirational surfaces in characteristic $p$, Math. Ann. 211 (1974), 233–236. MR 374149, DOI https://doi.org/10.1007/BF01350715

Tetsuji Shioda, On unirationality of supersingular surfaces, Math. Ann. 225 (1977), no. 2, 155–159. MR 429911, DOI https://doi.org/10.1007/BF01351719

Tetsuji Shioda, Some results on unirationality of algebraic surfaces, Math. Ann. 230 (1977), no. 2, 153–168. MR 572983, DOI https://doi.org/10.1007/BF01370660

Tetsuji Shioda, Supersingular $K3$ surfaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 564–591. MR 555718

Tetsuji Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240. MR 1081832

Tetsuji Shioda, Kummer sandwich theorem of certain elliptic $K3$ surfaces, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 137–140. MR 2279280

Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368

John Tate and Frans Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4) 3 (1970), 1–21. MR 265368

Additional Information

Hiroyuki Ito
Affiliation: Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan
Address at time of publication: Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan
Email: hiroito@amath.hiroshima-u.ac.jp\quad ito{_}hiroyuki@ma.noda.tus.ac.jp

Christian Liedtke
Affiliation: Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, California 94305-2125
Address at time of publication: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: liedtke@math.stanford.edu

Received by editor(s): April 16, 2010
Received by editor(s) in revised form: December 3, 2010, and January 3, 2011
Published electronically: March 14, 2012
Additional Notes: The first author acknowledges the support by Grant-in-Aid for Scientific Research (C) 20540044, the Ministry of Education, Culture, Sports, Science and Technology. The second author gratefully acknowledges funding from DFG under research grant LI 1906/1-1 and thanks the Department of Mathematics at Stanford University for kind hospitality.