Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T03:24:31.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic K3 Surfaces with Geometric Mordell–Weil Rank 15

Published online by Cambridge University Press:  20 November 2018

Remke Kloosterman
Remke Kloosterman*
Affiliation:
Institut für Algebraische Geometrie, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that the elliptic surface ${{y}^{2}}={{x}^{3}}+2\left( {{t}^{8}}+14{{t}^{4}}+1 \right)x+4{{t}^{2}}\left( {{t}^{8}}+6{{t}^{4}}+1 \right)$ has geometric Mordell–Weil rank 15. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell–Weil ranks 0, 1, … , 14, 16, 17, 18.

Keywords

14J2714J2811G05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 2 , 01 June 2007 , pp. 215 - 226
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Barth, W., Peters, C., and Van de Ven, A., Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer-Verlag, 1984.Google Scholar
[2] Barth, W. P., Hulek, K., Peters, C. A. M., and Van de Ven, A., Compact Complex Surfaces. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer-Verlag, Berlin, 2004.Google Scholar
[3] Cossec, F. R. and Dolgachev, I. V., Enriques Surfaces. I. Progress in Mathematics 76, Birkhäuser Boston, Boston, MA, 1989.Google Scholar
[4] Cox, D. A.,Mordell-Weil groups of elliptic curves over C(t) with pg = 0 or 1 . Duke Math. J. 49(1982), 677–689.Google Scholar
[5] Deligne, P., La conjecture de Weil. I. Inst. Hautes études Sci. Publ. Math. 43(1974), 273–307.Google Scholar
[6] Ellenberg, J. S., K3 surfaces over number fields with geometric Picard number one. In: Arithmetic of Higher-Dimensional Algebraic Varieties, Progr. Math. 226, Birkhäuser Boston, Boston, MA, 2004, pp. 135–140.Google Scholar
[7] Inose, H., On certain Kummer surfaces which can be realized as non-singular quartic surfaces in P 3 . J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23(1976), no. 3, 545–560.Google Scholar
[8] Kloosterman, R., Arithmetic and Moduli of Elliptic Surfaces. Ph.D. thesis, University of Groningen, 2005.Google Scholar
[9] Kloosterman, R., Higher Noether-Lefschetz loci for Elliptic Surfaces. Preprint available at arxiv:math.AG/0501454, 2005.Google Scholar
[10] Kuwata, M., Elliptic K 3 surfaces with given Mordell-Weil rank. Comment. Math. Univ. St. Paul. 49(2000), no. 1, 91–100.Google Scholar
[11] van Luijk, R.. An elliptic K3 surface associated to Heron triangles. Canad. Math. Bull. 49(2006), no. 4, 560–577.Google Scholar
[12] van Luijk, R., K3 surfaces with Picard number one and infinitely many rational points. Preprint available at arxiv:math.AG/0506416, 2005.Google Scholar
[13] Milne, J. S., On a conjecture of Artin and Tate. Ann. of Math. (2) 102(1975), no. 3, 517–533.Google Scholar
[14] Milne, J. S., Arithmetic duality theorems. Perspectives in Mathematics 1, Academic Press, Boston, MA, 1986.Google Scholar
[15] Miranda, R., The moduli of Weierstrass fibrations over P 1 . Math. Ann. 255(1981), 379–394.Google Scholar
[16] Miranda, R., The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica. ETS Editrice, Pisa, 1989.Google Scholar
[17] Nygaard, N. and Ogus, A., Tate's conjecture for K3 surfaces of finite height. Ann. of Math. (2) 122(1985), no. 3, 461–507.Google Scholar
[18] Oguiso, K. and Shioda, T., The Mordell-Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul. 40(1991), no. 1, 83–99.Google Scholar
[19] Shioda, T., On the Picard number of a complex projective variety. Ann. Sci. École Norm. Sup. (4) 14(1981), no. 3, 303–321.Google Scholar
[20] Shioda, T., On the Mordell-Weil lattices. Comment. Math. Univ. St. Paul. 39(1990), no. 2, 211–240.Google Scholar
[21] Silverman, J. H.. The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986.Google Scholar
[22] Silverman, J. H., Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 151, Springer-Verlag, New York, 1994.Google Scholar
[23] Tate, J., Algorithm for determining the type of a singular fibre in an elliptic pencil. In: Modular Functions of One Variable, IV, Lecture Notes in Mathematics 476, Springer-Verlag, Berlin, 1975, pp. 33–52.Google Scholar
[24] Tate, J., Conjectures on algebraic cycles in l-adic cohomology. In Motives, Proc. Sympos. Pure Math. 55, American Mathematical Society, Providence, RI, 1994, pp. 71–83.Google Scholar
[25] Terasoma, T., Complete intersections with middle Picard number 1 defined over Q . Math. Z. 189(1985), no. 2, 289–296.Google Scholar