On the uniqueness of elliptic K3 surfaces with maximal singular fibre

Matthias Schütt[1]; Andreas Schweizer[2]

  • [1] Institut für Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1 30167 Hannover Germany
  • [2] Department of Mathematics Korea Advanced Institute of Science and Technology (KAIST) Daejeon 305-701 South Korea

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 2, page 689-713
  • ISSN: 0373-0956

Abstract

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We explicitly determine the elliptic K 3 surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from 2 , for each of the two possible maximal fibre types, I 19 and I 14 * , the surface is unique. In characteristic 2 the maximal fibre types are I 18 and I 13 * , and there exist two (resp. one) one-parameter families of such surfaces.

How to cite

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Schütt, Matthias, and Schweizer, Andreas. "On the uniqueness of elliptic K3 surfaces with maximal singular fibre." Annales de l’institut Fourier 63.2 (2013): 689-713. <http://eudml.org/doc/275532>.

@article{Schütt2013,
abstract = {We explicitly determine the elliptic $K3$ surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from $2$, for each of the two possible maximal fibre types, $I_\{19\}$ and $I^*_\{14\}$, the surface is unique. In characteristic $2$ the maximal fibre types are $I_\{18\}$ and $I^*_\{13\}$, and there exist two (resp. one) one-parameter families of such surfaces.},
affiliation = {Institut für Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1 30167 Hannover Germany; Department of Mathematics Korea Advanced Institute of Science and Technology (KAIST) Daejeon 305-701 South Korea},
author = {Schütt, Matthias, Schweizer, Andreas},
journal = {Annales de l’institut Fourier},
keywords = {elliptic surface; $K3$ surface; maximal singular fibre; wild ramification; elliptic surfaces; surfaces; maximal singular fiber},
language = {eng},
number = {2},
pages = {689-713},
publisher = {Association des Annales de l’institut Fourier},
title = {On the uniqueness of elliptic K3 surfaces with maximal singular fibre},
url = {http://eudml.org/doc/275532},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Schütt, Matthias
AU - Schweizer, Andreas
TI - On the uniqueness of elliptic K3 surfaces with maximal singular fibre
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 2
SP - 689
EP - 713
AB - We explicitly determine the elliptic $K3$ surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from $2$, for each of the two possible maximal fibre types, $I_{19}$ and $I^*_{14}$, the surface is unique. In characteristic $2$ the maximal fibre types are $I_{18}$ and $I^*_{13}$, and there exist two (resp. one) one-parameter families of such surfaces.
LA - eng
KW - elliptic surface; $K3$ surface; maximal singular fibre; wild ramification; elliptic surfaces; surfaces; maximal singular fiber
UR - http://eudml.org/doc/275532
ER -

References

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  1. M. Artin, Supersingular K 3 surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543-568 Zbl0322.14014MR371899
  2. M. Artin, P. Swinnerton-Dyer, The Shafarevich-Tate conjecture for pencils of elliptic curves on K 3 surfaces, Invent. Math. 20 (1973), 249-266 Zbl0289.14003MR417182
  3. F. Beukers, H. Montanus, Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps, Number theory and polynomials 352 (2008), 33-51, Cambridge Univ. Press, Cambridge Zbl1266.11078MR2428514
  4. F. R. Cossec, I. V. Dolgachev, Enriques surfaces, 76 (1989), Birkhäuser Zbl0665.14017MR986969
  5. I. Dolgachev, S. Kondō, A supersingular K3 surface in characteristic 2 and the Leech lattice, Int. Math. Res. Not. (2003), 1-23 Zbl1061.14031MR1935564
  6. N. D. Elkies, Mordell-Weil lattices in characteristic 2. I. Construction and first properties, Int. Math. Res. Not. (1994), 343-361 Zbl0813.52017MR1289579
  7. E.-U. Gekeler, Local and global ramification properties of elliptic curves in characteristics two and three, Algorithmic Algebra and Number Theory (1998), 49-64, MatzatB. H.B. H. Zbl1099.11507MR1672097
  8. M. Hall Jr., The Diophantine equation x 3 - y 2 = k , 173-198 
  9. K. Kodaira, On compact analytic surfaces II, III, Ann. of Math. (2) 77 (1963), 563-626 Zbl0118.15802MR184257
  10. R. Livné, Motivic Orthogonal Two-dimensional Representations of Gal ( ℚ ¯ / ℚ ) , Israel J. Math. 92 (1995), 149-156 Zbl0847.11035MR1357749
  11. R. Miranda, U. Persson, Configurations of I n Fibers on Elliptic K3 surfaces, Math. Z. 201 (1989), 339-361 Zbl0694.14019MR999732
  12. R. Miyamoto, J. Top, Reduction of Elliptic Curves in Equal Characteristic, Canad. Math. Bull. 48 (2005), 428-444 Zbl1100.11018MR2154085
  13. J. Pesenti, L. Szpiro, Inégalité du discriminant pour les pinceaux elliptiques à réductions quelconques, Compositio Math. 120 (2000), 83-117 Zbl1021.11021MR1738213
  14. I. I. Pjateckiĭ-Šapiro, I. R Šafarevič, A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv. 5 (1972), 547-588 Zbl0253.14006
  15. A. Schweizer, Extremal elliptic surfaces in characteristic 2 and 3 , Manuscripta Math. 102 (2000), 505-521 Zbl0989.14013MR1785328
  16. M. Schütt, The maximal singular fibres of elliptic K3 surfaces, Arch. Math. (Basel) 87 (2006), 309-319 Zbl1111.14034MR2263477
  17. M. Schütt, CM newforms with rational coefficients, Ramanujan J. 19 (2009), 187-205 Zbl1226.11057MR2511671
  18. M. Schütt, A. Schweizer, Davenport-Stothers inequalities and elliptic surfaces in positive characteristic, Quarterly J. Math. 59 (2008), 499-522 Zbl1154.14028MR2461271
  19. M. Schütt, J. Top, Arithmetic of the [19,1,1,1,1,1] fibration, Comm. Math. Univ. St. Pauli 55 (2006), 9-16 Zbl1125.14022MR2251996
  20. T. Shioda, On the Mordell-Weil lattices, Comm. Math. Univ. St. Pauli 39 (1990), 211-240 Zbl0725.14017MR1081832
  21. T. Shioda, The elliptic K3 surfaces with a maximal singular fibre, C. R. Acad. Sci. Paris, Ser. I 337 (2003), 461-466 Zbl1048.14017MR2023754
  22. T. Shioda, Elliptic surfaces and Davenport-Stothers triples, Comment. Math. Univ. St. Pauli 54 (2005), 49-68 Zbl1100.14031MR2153955
  23. T. Shioda, H. Inose, On Singular K 3 Surfaces, Baily W. L. Jr., Shioda, T. (eds.) (1977), 119-136, Iwanami Shoten, Tokyo Zbl0374.14006MR441982
  24. J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, (1994), Springer GTM, Berlin-Heidelberg-New York Zbl0911.14015MR1312368
  25. W. W. Stothers, Polynomial identities and Hauptmoduln, Quart. J. Math. Oxford (2) 32 (1981), 349-370 Zbl0466.12011MR625647
  26. J. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (1965), 93-110, Harper & Row Zbl0213.22804MR225778
  27. J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil, Modular functions of one variable IV 476 (1975), 33-52 Zbl1214.14020MR393039

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