Hostname: page-component-797576ffbb-vjhkx Total loading time: 0 Render date: 2023-12-09T03:38:40.210Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false

On the computation of the Picard group for K3 surfaces

Published online by Cambridge University Press:  10 June 2011

Mathematisches Institut, Universität Bayreuth, Universitätsstraße 30, D-95440 Bayreuth, Germany. e-mail:
Fachbereich 6 Mathematik, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany. e-mail:


We present a method to construct examples of K3 surfaces of geometric Picard rank 1. Our approach is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on étale cohomology. This allows us to abandon the original limitation to cases of Picard rank 2 after reduction modulo p. Furthermore, the use of Galois data enables us to construct examples that require significantly less computation time.

Research Article
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



[1]Barth, W., Peters, C. and Van de Ven, A.. Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 4 (Springer, 1984).Google Scholar
[2]Beauville, A.. Surfaces algébriques complexes. Astérisque 54. (Société Mathématique de France, Paris, 1978).Google Scholar
[3]Deligne, P.. La conjecture de Weil I. Publ. Math. Inst. Hawks Études Sci 43 (1974), 273307.Google Scholar
[4]Elsenhans, A.-S. and Jahnel, J.. K3 surfaces of Picard rank one and degree two, in: Algorithmic Number Theory (ANTS 8) 212225 (Springer, 2008).Google Scholar
[5]Elsenhans, A.-S. and Jahnel, J.. The discriminant of a cubic surface. Preprint, Scholar
[6]Elsenhans, A.-S. and Jahnel, J.. Cubic surfaces with a Galois invariant double-six. Central European Journal of Mathematics 8 (2010), 646661.Google Scholar
[7]Elsenhans, A.-S. and Jahnel, J.. Cubic surfaces with a Galois invariant pair of Steiner trihedra, to appear in: International Journal of Number TheoryGoogle Scholar
[8]Fulton, W.. Intersection Theory (Springer, 1984).Google Scholar
[9]Kloosterman, R.. Elliptic K3 surfaces with geometric Mordell–Weil rank 15. Canad. Math. Bull. 50 (2007), 215226.Google Scholar
[10]Lieberman, D. I.. Numerical and homological equivalence of algebraic cycles on Hodge manifolds. Amer. J. Math. 90 (1968), 366374.Google Scholar
[11]Liu, Q., Lorenzini, D. and Raynaud, M.. On the Brauer group of a surface. Invent. Math. 159 (2005), 673676.Google Scholar
[12]van Luijk, R.. K3 surfaces with Picard number one and infinitely many rational points. Algebra Number Theory 1 (2007), 115.Google Scholar
[13]Milne, J. S.. Étale Cohomology (Princeton University Press, 1980).Google Scholar
[14]Milne, J. S.. On a conjecture of Artin and Tate. Ann. of Math. 102 (1975), 517533.Google Scholar
[15]Pohst, M. and Zassenhaus, H.. Algorithmic Algebraic Number Theory, (Cambridge University Press, 1989).Google Scholar
[16]Tate, J.. Conjectures on algebraic cycles in l-adic cohomology, in: Motives. Proc. Sympos. Pure Math. 55-1 (Amer. Math. Soc., Providence 1994), 7183.Google Scholar