Abstract
We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface X whose transcendental lattice is isometric to \(\langle 6\rangle \oplus \langle 2\rangle\).
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References
Atkin, A.O., Morain, F.: Finding suitable curves for the elliptic curve method of factorization. Math. Comput. 60, 399–405 (1993)
Beauville, A.: Les familles stables de courbes elliptiques sur \(\mathbb{P}^{1}\) admettant quatre fibres singulières. C. R. Acad. Sci. Paris Sér. I Math. 294, 657–660 (1982)
Bertin, M.J.: Mesure de Mahler et série L d’une surface K3 singulière. Actes de la Conférence: Fonctions L et Arithmétique. Publ. Math. Besan. Actes de la conférence Algèbre Théorie Nbr., Lab. Math. Besançon, pp. 5–28 (2010)
Bertin, M.J., Lecacheux, O.: Elliptic fibrations on the modular surface associated to \(\Gamma _{1}(8)\). In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Fields Institute Communications, vol. 67, pp. 153–199. Springer, New York (2013)
Braun, A.P., Kimura, Y., Watari, T.: On the classification of elliptic fibrations modulo isomorphism on K3 surfaces with large Picard number, Math. AG; High Energy Physics (2013) [arXiv:1312.4421]
Cassels, J.W.S.: Lectures on Elliptic Curves. London Mathematical Society Student Texts, vol. 24. Cambridge University Press, Cambridge (1991)
Comparin, P., Garbagnati, A.: Van Geemen-Sarti involutions and elliptic fibrations on K3 surfaces double cover of \(\mathbb{P}^{2}\). J. Math. Soc. Jpn. 66, 479–522 (2014)
Couveignes, J.-M., Edixhoven, S.: Computational Aspects of Modular Forms and Galois Representations, Annals of Math Studies 176. Princeton University Press, Princeton
Cox, D., Katz, S.: Mirror Symmetry and Algebraic Geometry. American Mathematical Society, Providence (1999)
Elkies, N.D.: http://www.math.harvard.edu/~elkies/K3_20SI.html#-7[1 (2010)
Elkies, N.D.: Three Lectures on Elliptic Surfaces and Curves of High Rank. Lecture notes, Oberwolfach (2007)
Elkies, N., Schütt, M.: Genus 1 fibrations on the supersingular K3 surface in characteristic 2 with Artin invariant 1. Asian J. Math. (2014) [arXiv:1207.1239]
Garbagnati, A., Sarti, A.: Elliptic fibrations and symplectic automorphisms on K3 surfaces. Commun. Algebra 37, 3601–3631 (2009)
Karp, D., Lewis, J., Moore, D., Skjorshammer, D., Whitcher, U.: On a family of K3 surfaces with \(\mathcal{S}_{4}\) symmetry. In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Fields Institute Communications. Springer, New York (2013)
Kloosterman, R.: Classification of all Jacobian elliptic fibrations on certain K3 surfaces. J. Math. Soc. Jpn. 58, 665–680 (2006)
Kondo, S.: Algebraic K3 surfaces with finite automorphism group. Nagoya Math. J. 116, 1–15 (1989)
Kondo, S.: Automorphisms of algebraic K3 surfaces which act trivially on Picard groups. J. Math. Soc. Jpn. 44, 75–98 (1992)
Kubert, D.S.: Universal bounds on the torsion of elliptic curves. Proc. Lond. Math. Soc. 33(3), 193–237 (1976)
Kumar, A.: Elliptic fibrations on a generic Jacobian Kummer surface. J. Algebraic Geom. [arXiv:1105.1715] 23, 599–667 (2014)
Kreuzer, M., Skarke, H.: Classification of reflexive polyhedra in three dimensions. Adv. Theor. Math. Phys. 2, 853–871 (1998)
Kreuzer, M., Skarke, H.: Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4, 1209–1230 (2000)
Martinet, J.: Perfect Lattices in Euclidean Spaces, vol. 327. Springer, Berlin/Heidelberg (2002)
Montgomery, P.L.: Speeding the Pollard and Elliptic Curve Methods of Factorization. Math. Comput. 48. (1987) 243–264.
Nikulin, V.V.: Finite groups of automorphisms of Kählerian K3 surfaces. (Russian) Trudy Moskov. Mat. Obshch. 38, 75–137 (1979). English translation: Trans. Moscow Math. Soc. 38, 71–135 (1980)
Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izv. 14, 103–167 (1980)
Nishiyama, K.: The Jacobian fibrations on some K3 surfaces and their Mordell–Weil groups. Jpn. J. Math. (N.S.) 22, 293–347 (1996)
Oguiso, K.: On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves. J. Math. Soc. Jpn. 41, 651–680 (1989)
Schütt, M.: K3 surface with Picard rank 20 over \(\mathbb{Q}\). Algebra Number Theory 4, 335–356 (2010)
Schütt, M., Shioda, T.: Elliptic surfaces. In: Algebraic Geometry in East Asia–Seoul 2008. Advanced Studies in Pure Mathematics, vol. 60, pp. 51–160. The Mathematical Society of Japan, Tokyo (2010)
Shimada, I., Zhang, D.-Q.: Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces. Nagoya Math. J. 161, 23–54 (2001)
Shioda, T.: On the Mordell–Weil lattices. Commun. Math. Univ. St. Pauli 39, 211–240 (1990)
Shioda, T.: On elliptic modular surfaces J. Math. Soc. Jpn. 24(1), 20–59 (1972)
Stein, W.A., et al.: Sage Mathematics Software (Version 6.1). The Sage Development Team. http://www.sagemath.org (2014)
Sterk, H.: Finiteness results for algebraic K3 surfaces. Math. Z. 180, 507–513 (1985)
Verrill, H.: Root lattices and pencils of varieties. J. Math. Kyoto Univ. 36, 423–446 (1996)
Acknowledgements
We thank the organizers and all those who supported our project for their efficiency, their tenacity and expertise. The authors of the paper have enjoyed the hospitality of CIRM at Luminy, which helped to initiate a very fruitful collaboration, gathering from all over the world junior and senior women, bringing their skill, experience, and knowledge from geometry and number theory. Our gratitude goes also to the referee for pertinent remarks and helpful comments.
A.G is supported by FIRB 2012 “Moduli Spaces and Their Applications” and by PRIN 2010–2011 “Geometria delle varietà algebriche.” C.S is supported by FAPERJ (grant E26/112.422/2012). U.W. thanks the NSF-AWM Travel Grant Program for supporting her visit to CIRM.
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Bertin, M.J. et al. (2015). Classifications of Elliptic Fibrations of a Singular K3 Surface. In: Bertin, M., Bucur, A., Feigon, B., Schneps, L. (eds) Women in Numbers Europe. Association for Women in Mathematics Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-17987-2_2
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DOI: https://doi.org/10.1007/978-3-319-17987-2_2
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