Hostname: page-component-7dd5485656-npwhs Total loading time: 0 Render date: 2025-10-29T13:24:05.439Z Has data issue: false hasContentIssue false
  • English
  • Français

Examples of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K3$ surfaces with real multiplication

Part of: Arithmetic algebraic geometry Cycles and subschemes Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  01 August 2014

Andreas-Stephan Elsenhans  and
Jörg Jahnel
Andreas-Stephan Elsenhans
Affiliation:
School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia email stephan@maths.usyd.edu.au
Jörg Jahnel
Affiliation:
Département Mathematik, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany email jahnel@mathematik.uni-siegen.de

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 construct explicit $K3$ surfaces over $\mathbb{Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces constructed.

MSC classification

Secondary: 14J28: $K3$ surfaces and Enriques surfaces 14C30: Transcendental methods, Hodge theory , Hodge conjecture 11G15: Complex multiplication and moduli of abelian varieties 14C22: Picard groups

References

Barth, W., Peters, C. and van de Ven, A., Compact complex surfaces (Springer, Berlin, Heidelberg, New York, Tokyo, 2004).Google Scholar
Beauville, A., ‘Surfaces algébriques complexes’, Astérisque 54 (Société Mathématique de France, Paris, 1978).Google Scholar
Berthelot, P. and Ogus, A., Notes on crystalline cohomology (Princeton University Press, Princeton, 1978).Google Scholar
Charles, F., ‘On the Picard number of K3 surfaces over number fields’, Algebra Number Theory 8 (2014) 1–17.CrossRefGoogle Scholar
Charles, F., ‘The Tate conjecture for K3 surfaces over finite fields’, Invent. Math. 194 (2013) 119–145.Google Scholar
Cox, D. A., Primes of the form x 2 + n y 2: Fermat, class field theory and complex multiplication (John Wiley & Sons, New York, 1989).Google Scholar
Deligne, P., ‘Théorie de Hodge II’, Publ. Math. Inst. Hautes Études Sci. 40 (1971) 5–57.CrossRefGoogle Scholar
Deligne, P., ‘La conjecture de Weil I’, Publ. Math. Inst. Hautes Études Sci. 43 (1974) 273–307.Google Scholar
Dixon, J. D. and Mortimer, B., Permutation groups, Graduate Texts in Mathematics 163 (Springer, New York, 1996).CrossRefGoogle Scholar
Elkies, N. and Kumar, A., ‘$K3$ surfaces and equations for Hilbert modular surfaces’, Preprint, 2012, arXiv:1209.3527.Google Scholar
Elsenhans, A.-S. and Jahnel, J., ‘K3 surfaces of Picard rank one and degree two’, Algorithmic number theory (ANTS 8), Lecture Notes in Computer Science 5011 (Springer, Berlin, 2008) 212–225.CrossRefGoogle Scholar
Elsenhans, A.-S. and Jahnel, J., ‘On Weil polynomials of K3 surfaces’, Algorithmic number theory (ANTS 9), Lecture Notes in Computer Science 6197 (Springer, Berlin, 2010) 126–141.CrossRefGoogle Scholar
Elsenhans, A.-S. and Jahnel, J., ‘Kummer surfaces and the computation of the Picard group’, LMS J. Comput. Math. 15 (2012) 84–100.CrossRefGoogle Scholar
Elsenhans, A.-S. and Jahnel, J., ‘On the computation of the Picard group for certain singular quartic surfaces’, Math. Slovaca 63 (2013) 215–228.Google Scholar
Faltings, G., ‘p-adic Hodge theory’, J. Amer. Math. Soc. 1 (1988) 255–299.Google Scholar
Fisher, T., ‘The invariants of a genus one curve’, Proc. Lond. Math. Soc. 97 (2008) 753–782.CrossRefGoogle Scholar
Fité, F., Kedlaya, K., Rotger, V. and Sutherland, A. V., ‘Sato–Tate distributions and Galois endomorphism modules in genus 2’, Compos. Math. 148 (2012) 1390–1442.CrossRefGoogle Scholar
Grothendieck, A., Fondements de la Géométrie Algébrique (FGA), Séminaire Bourbaki 149, 182, 190, 195, 212, 221, 232, 236 (Paris, 1957–62).Google Scholar
van Geemen, B., ‘Real multiplication on K3 surfaces and Kuga-Satake varieties’, Michigan Math. J. 56 (2008) 375–399.CrossRefGoogle Scholar
Hall, M. Jr., Combinatorial theory (Blaisdell Publishing Co., Waltham, Toronto, London, 1967).Google Scholar
Harris, M., Shepherd-Barron, N. and Taylor, R., ‘A family of Calabi-Yau varieties and potential automorphy’, Ann. of Math. 171 (2010) 779–813.CrossRefGoogle Scholar
Humbert, G., ‘Sur les fonctions abéliennes singulières’, J. Math. Pures Appl. 5e série 5 (1899) 233–350.Google Scholar
Illusie, L., ‘Crystalline cohomology’, Motives (Seattle 1991), Proceedings of Symposia in Pure Mathematics 55-1 (American Mathematical Society, Providence, RI, 1994) 43–70.Google Scholar
Illusie, L., ‘Perversité et variation’, Manuscripta Math. 112 (2003) 271–295.CrossRefGoogle Scholar
Illusie, L. and Raynaud, M., ‘Les suites spectrales associées au complexe de de Rham–Witt’, Publ. Math. Inst. Hautes Études Sci. 57 (1983) 73–212.CrossRefGoogle Scholar
Katz, N. M. and Mazur, B., Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108 (Princeton University Press, Princeton, 1985).Google Scholar
Kedlaya, K. S. and Sutherland, A. V., ‘Hyperelliptic curves, L-polynomials, and random matrices’, Arithmetic, geometry, cryptography and coding theory, Contemporary Mathematics 487 (American Mathematical Society, Providence, 2009) 119–162.CrossRefGoogle Scholar
Larsen, M. and Pink, R., ‘On l-independence of algebraic monodromy groups in compatible systems of representations’, Invent. Math. 107 (1992) 603–636.CrossRefGoogle Scholar
Lieblich, M., Maulik, D. and Snowden, A., ‘Finiteness of $K3$ surfaces and the Tate conjecture’, Preprint, 2011, arXiv:1107.1221.Google Scholar
Liedtke, C., ‘Lectures on supersingular $K3$ surfaces and the crystalline Torelli theorem’, Preprint, 2014, arXiv:1403.2538.Google Scholar
van Luijk, R., ‘Rational points on $K3$ surfaces’, PhD Thesis, Berkeley, 2005.Google Scholar
van Luijk, R., ‘K3 surfaces with Picard number one and infinitely many rational points’, Algebra Number Theory 1 (2007) 1–15.CrossRefGoogle Scholar
Mazur, B., ‘Frobenius and the Hodge filtration (estimates)’, Ann. of Math. (2) 98 (1973) 58–95.CrossRefGoogle Scholar
Milne, J. S., ‘On a conjecture of Artin and Tate’, Ann. of Math. (2) 102 (1975) 517–533.CrossRefGoogle Scholar
Neukirch, J., Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322 (Springer, Berlin, 1999).CrossRefGoogle Scholar
Ochiai, T., ‘l-independence of the trace of monodromy’, Math. Ann. 315 (1999) 321–340.CrossRefGoogle Scholar
Pera, K. M., ‘The Tate conjecture for $K3$ surfaces in odd characteristic’, Preprint, 2013, arXiv:1301.6326.Google Scholar
Serre, J.-P., Cours d’arithmétique (Presses Universitaires de France, Paris, 1970).Google Scholar
Artin, M., Grothendieck, A. and Verdier, J.-L., Théorie des Topos et Cohomologie Étale des Schémas (Séminaire de Géométrie Algébrique du Bois Marie 1963–1964 (SGA 4)), Lecture Notes in Mathematics 269, 270, 305 (Springer, 1972–1973).Google Scholar
Grothendieck, A., Cohomologie l-adique et Fonctions L (Séminaire de Géométrie Algébrique du Bois Marie 1965–1966 (SGA 5)), Lecture Notes in Mathematics 589 (Springer, 1977).Google Scholar
Deligne, P. and Katz, N., Groupes de Monodromie en Géométrie Algébrique, Séminaire de Géométrie Algébrique du Bois Marie 1967–1969 (SGA 7), Lecture Notes in Mathematics 288, 340 (Springer, 1973).Google Scholar
Petrovskiǐ, I. G. and Nikol’skiǐ, S. M. (eds), Algebraic surfaces, Proceedings of the Steklov Institute of Mathematics 75 (American Mathematical Society, Providence, RI, 1967).Google Scholar
Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151 (Springer, New York, 1994).CrossRefGoogle Scholar
Tankeev, S. G., ‘Surfaces of K3 type over number fields and the Mumford–Tate conjecture’, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990) 846–861 (in Russian).Google Scholar
Tankeev, S. G., ‘Surfaces of K3 type over number fields and the Mumford–Tate conjecture II’, Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995) 179–206 (in Russian).Google Scholar
Weber, H., Lehrbuch der Algebra 2, Auflage, 3. Band: Elliptische Funktionen und algebraische Zahlen (Friedr. Vieweg & Sohn, Braunschweig, 1908).Google Scholar
Zarhin, Yu. G., ‘Hodge groups of K3 surfaces’, J. reine angew. Math. 341 (1983) 193–220.Google Scholar
Zarhin, Yu. G., ‘Transcendental cycles on ordinary K3 surfaces over finite fields’, Duke Math. J. 72 (1993) 65–83.CrossRefGoogle Scholar