On the uniqueness of elliptic K3 surfaces with maximal singular fibre
[Sur l’unicité des surfaces elliptiques de type K3 ayant une fibre singulière maximale]
Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 689-713.

Nous déterminons des équations explicites pour les surfaces elliptiques de type K3 qui ont une section et une fibre singulière maximale. Si la caractéristique du corps sous-jacent est différente de 2, pour chacun des deux types de fibre maximale, I 19 et I 14 * , la surface est unique. En caractéristique 2 les fibres maximales sont de type I 18 ou I 13 * , et il y a deux, respectivement une, familles 1-dimensionales de telles surfaces.

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.

DOI : 10.5802/aif.2773
Classification : 14J27, 14J28, 11G05
Keywords: elliptic surface, $K3$ surface, maximal singular fibre, wild ramification.
Mot clés : surface elliptique, surface de type $K3$, fibre singulière maximale, ramification sauvage
Schütt, Matthias 1 ; Schweizer, Andreas 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
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Schütt, Matthias; Schweizer, Andreas. On the uniqueness of elliptic K3 surfaces with maximal singular fibre. Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 689-713. doi : 10.5802/aif.2773. https://aif.centre-mersenne.org/articles/10.5802/aif.2773/

[1] Artin, M. Supersingular K3 surfaces, Ann. Sci. École Norm. Sup. (4), Volume 7 (1974), pp. 543-568 | Numdam | MR | Zbl

[2] Artin, M.; Swinnerton-Dyer, P. The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math., Volume 20 (1973), pp. 249-266 | DOI | MR | Zbl

[3] Beukers, F.; Montanus, H. Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps, Number theory and polynomials (London Math. Soc. Lecture Note Ser.), Volume 352, Cambridge Univ. Press, Cambridge, 2008, pp. 33-51 | MR

[4] Cossec, F. R.; Dolgachev, I. V. Enriques surfaces, I. Progress in Math., 76, Birkhäuser, 1989 | MR | Zbl

[5] Dolgachev, I.; Kondō, S. A supersingular K3 surface in characteristic 2 and the Leech lattice, Int. Math. Res. Not. (2003) no. 1, pp. 1-23 | DOI | MR | Zbl

[6] Elkies, N. D. Mordell-Weil lattices in characteristic 2. I. Construction and first properties, Int. Math. Res. Not. (1994) no. 8, pp. 343-361 | DOI | MR | Zbl

[7] Gekeler, E.-U.; Matzat, B. H.; Greuel, G.-M.; Hiß, G. Local and global ramification properties of elliptic curves in characteristics two and three, Algorithmic Algebra and Number Theory, Berlin-Heidelberg-New York: Springer, 1998, pp. 49-64 | MR | Zbl

[8] Hall Jr., M. The Diophantine equation x 3 -y 2 =k (Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969)), pp. 173-198

[9] Kodaira, K. On compact analytic surfaces II, III, Ann. of Math. (2), Volume 77 (1963), pp. 563-626 ibid. 78 (1963), 1–40 | DOI | MR | Zbl

[10] Livné, R. Motivic Orthogonal Two-dimensional Representations of Gal( ¯/), Israel J. Math., Volume 92 (1995), pp. 149-156 | DOI | MR | Zbl

[11] Miranda, R.; Persson, U. Configurations of I n Fibers on Elliptic K3 surfaces, Math. Z., Volume 201 (1989), pp. 339-361 | DOI | MR | Zbl

[12] Miyamoto, R.; Top, J. Reduction of Elliptic Curves in Equal Characteristic, Canad. Math. Bull., Volume 48 (2005), pp. 428-444 | DOI | MR | Zbl

[13] Pesenti, J.; Szpiro, L. Inégalité du discriminant pour les pinceaux elliptiques à réductions quelconques, Compositio Math., Volume 120 (2000) no. 1, pp. 83-117 | DOI | MR | Zbl

[14] Pjateckiĭ-Šapiro, I. I.; Šafarevič, I. R A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv., Volume 5 (1972) no. 3, pp. 547-588 | DOI | Zbl

[15] Schweizer, A. Extremal elliptic surfaces in characteristic 2 and 3, Manuscripta Math., Volume 102 (2000), pp. 505-521 | DOI | MR | Zbl

[16] Schütt, M. The maximal singular fibres of elliptic K3 surfaces, Arch. Math. (Basel), Volume 87 (2006) no. 4, pp. 309-319 | DOI | MR | Zbl

[17] Schütt, M. CM newforms with rational coefficients, Ramanujan J., Volume 19 (2009), pp. 187-205 | DOI | MR | Zbl

[18] Schütt, M.; Schweizer, A. Davenport-Stothers inequalities and elliptic surfaces in positive characteristic, Quarterly J. Math., Volume 59 (2008), pp. 499-522 | DOI | MR | Zbl

[19] Schütt, M.; Top, J. Arithmetic of the [19,1,1,1,1,1] fibration, Comm. Math. Univ. St. Pauli, Volume 55 (2006) no. 1, pp. 9-16 | MR | Zbl

[20] Shioda, T. On the Mordell-Weil lattices, Comm. Math. Univ. St. Pauli, Volume 39 (1990), pp. 211-240 | MR | Zbl

[21] Shioda, T. The elliptic K3 surfaces with a maximal singular fibre, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 461-466 | DOI | MR | Zbl

[22] Shioda, T. Elliptic surfaces and Davenport-Stothers triples, Comment. Math. Univ. St. Pauli, Volume 54 (2005), pp. 49-68 | MR | Zbl

[23] Shioda, T.; Inose, H. On Singular K3 Surfaces, Baily W. L. Jr., Shioda, T. (eds.) (Complex analysis and algebraic geometry), Iwanami Shoten, Tokyo, 1977, pp. 119-136 | MR | Zbl

[24] Silverman, J. H. Advanced Topics in the Arithmetic of Elliptic Curves, Springer GTM, Berlin-Heidelberg-New York, 1994 | MR | Zbl

[25] Stothers, W. W. Polynomial identities and Hauptmoduln, Quart. J. Math. Oxford (2), Volume 32 (1981), pp. 349-370 | DOI | MR | Zbl

[26] Tate, J. Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry ((Proc. Conf. Purdue Univ., 1963)), Harper & Row, 1965, pp. 93-110 | MR | Zbl

[27] Tate, J. Algorithm for determining the type of a singular fibre in an elliptic pencil, Modular functions of one variable IV ((Antwerpen 1972), SLN), Volume 476, 1975, pp. 33-52 | MR | Zbl

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