Algebraic Geometry
The elliptic K3 surfaces with a maximal singular fibre
Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 461-466.

We give the defining equation of complex elliptic K3 surfaces with a maximal singular fibre. Then we study the reduction modulo p at a particularly interesting prime p.

Nous donnons l'équation des surfaces K3 elliptiques possédant une fibre singulière maximale. Puis nous étudions leur réduction modulo p, où p est un nombre premier particulièrement intéressant.

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DOI: 10.1016/j.crma.2003.07.007
Shioda, Tetsuji 1

1 Department of Mathematics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
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Shioda, Tetsuji. The elliptic K3 surfaces with a maximal singular fibre. Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 461-466. doi : 10.1016/j.crma.2003.07.007. http://www.numdam.org/articles/10.1016/j.crma.2003.07.007/

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

[2] Beauville, A. Les familles stables de courbes elliptiques sur P 1 , C. R. Acad. Sci. Paris, Ser. I, Volume 294 (1982), pp. 657-660

[3] Birch, B.J.; Chowla, S.; Hall, M.; Schinzel, A. On the difference x3y2, Norske Vid. Selsk. Forh. (Trondheim), Volume 38 (1965), pp. 65-69

[4] Davenport, H. On f3(t)−g2(t), Norske Vid. Selsk. Forh. (Trondheim), Volume 38 (1965), pp. 86-87

[5] Hall, M. The Diophantine equation x3y2=k, Computers in Number Theory, Academic Press, 1971, pp. 173-198

[6] Hellegouarch, Y. Analogues en caractéristique p d'un théoreme de Mason, C. R. Acad. Sci. Paris, Ser. I, Volume 325 (1997), pp. 141-144

[7] Inose, H.; Shioda, T. ‘On singular K3 surfaces, Complex Analysis and Algebraic Geometry, Iwanami Shoten and Cambridge Univ. Press, 1977, pp. 119-136

[8] Ito, H. On extremal elliptic surfaces in characteristic 2 and 3, Hiroshima Math. J., Volume 32 (2002), pp. 179-188

[9] Kodaira, K., Ann. of Math. (Collected Works), Volume 77, Iwanami and Princeton University Press, 1963, pp. 563-626 78 (1963) 1–40

[10] Miranda, R.; Persson, U. Configurations of In fibers on elliptic K3 surfaces, Math. Z., Volume 201 (1989), pp. 339-361

[11] Nishiyama, K. The Jacobian fibrations on some K3 surfaces and their Mordell–Weil groups, Japan. J. Math., Volume 22 (1996), pp. 293-347

[12] Oguiso, K.; Shioda, T. The Mordell–Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli, Volume 40 (1991), pp. 83-99

[13] U. Schmickler-Hirzebruch, Elliptische Fläche über P 1 C mit drei Ausnahmefasern und die Hypergeometrische Differentialgleichung, Diplomarbeit, Univ. Bonn, 1978

[14] I. Shimada, Rational double points on supersingular K3 surfaces, Preprint

[15] Shioda, T. On elliptic modular surfaces, J. Math. Soc. Japan, Volume 24 (1972), pp. 20-59

[16] Shioda, T. On the Mordell–Weil lattices, Comment. Math. Univ. St. Pauli, Volume 39 (1990), pp. 211-240

[17] Shioda, T. Integral points and Mordell–Weil lattices, A Panorama in Number Theory or The View from Baker's Garden, Cambridge University Press, 2002, pp. 185-193

[18] T. Shioda, Elliptic surfaces and Davenport–Stothers triples, in preparation

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

[20] Tate, J. Algorithm for determining the type of a singular fiber in an elliptic pencil, SLN, Volume 476 (1975), pp. 33-52

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