Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T05:20:26.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing Néron–Severi groups and cycle class groups

Part of: Arithmetic problems. Diophantine geometry Cycles and subschemes (Co)homology theory

Published online by Cambridge University Press:  04 February 2015

Bjorn Poonen ,
Damiano Testa  and
Ronald van Luijk
Bjorn Poonen
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA email [email protected]
Damiano Testa
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email [email protected]
Ronald van Luijk
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA, Leiden, The Netherlands email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Assuming the Tate conjecture and the computability of étale cohomology with finite coefficients, we give an algorithm that computes the Néron–Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension $p$ cycles for any $p$.

Keywords

Néron–Severi groupscycle class groupsTate conjecture

MSC classification

Primary: 14C22: Picard groups
Secondary: 14C25: Algebraic cycles 14F20: Étale and other Grothendieck topologies and (co)homologies 14G25: Global ground fields 14G27: Other nonalgebraically closed ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 4 , April 2015 , pp. 713 - 734
Copyright
© The Author(s) 2015 

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.)

References

André, Y., On the Shafarevich and Tate conjectures for hyper-Kähler varieties, Math. Ann. 305 (1996), 205–248; doi:10.1007/BF01444219, MR 1391213 (97a:14010).Google Scholar
Artin, M., On the joins of Hensel rings, Adv. Math. 7 (1971), 282–296; MR 0289501 (44 #6690).CrossRefGoogle Scholar
Basu, S., Pollack, R. and Roy, M.-F., Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics, vol. 10, second edition (Springer, Berlin, 2006); MR 2248869 (2007b:14125).Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21 (Springer, Berlin, 1990); MR 1045822 (91i:14034).Google Scholar
Charles, F., The Tate conjecture for K3 surfaces over finite fields, Invent. Math. 194 (2013), 119–145; doi:10.1007/s00222-012-0443-y, MR 3103257.Google Scholar
Charles, F., On the Picard number of K3 surfaces over number fields, Algebra Number Theory 8 (2014), 1–17; doi:10.2140/ant.2014.8.1, MR 3207577.CrossRefGoogle Scholar
Deligne, P., Relèvement des surfaces K3 en caractéristique nulle, in Algebraic surfaces, Lecture Notes in Mathematics, vol. 868 (Springer, Berlin, 1981), 58–79 (in French). Prepared for publication by Luc Illusie; MR 638598 (83j:14034).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci. 32 (1967), 361 (in French);MR 0238860 (39 #220).Google Scholar
Eisenbud, D., Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer, New York, 1995); MR 1322960 (97a:13001).Google Scholar
Elsenhans, A.-S. and Jahnel, J., The Picard group of a K3 surface and its reduction modulo p, Algebra Number Theory 5 (2011), 1027–1040.Google Scholar
Elsenhans, A.-S. and Jahnel, J., On the computation of the Picard group for K3 surfaces, Math. Proc. Cambridge Philos. Soc. 151 (2011), 263–270; doi:10.1017/S0305004111000326,MR 2823134 (2012i:14015).Google Scholar
Gotzmann, G., Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z. 158 (1978), 61–70 (in German); MR 0480478 (58 #641).Google Scholar
Grothendieck, A., Standard conjectures on algebraic cycles, in Algebraic geometry (Internat. Colloq. Tata Inst. Fund. Res. Bombay, 1968) (Oxford University Press, London, 1969), 193–199; MR 0268189 (42 #3088).Google Scholar
Haran, D., Quantifier elimination in separably closed fields of finite imperfectness degree, J. Symbolic Logic 53 (1988), 463–469; doi:10.2307/2274518, MR 947853 (89i:03057).Google Scholar
Hassett, B., Kresch, A. and Tschinkel, Y., Effective computation of Picard groups and Brauer–Manin obstructions of degree two K3 surfaces over number fields, Rend. Circ. Mat. Palermo (2) 62 (2013), 137–151; doi:10.1007/s12215-013-0116-8, MR 3031574.Google Scholar
Hermann, G., Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), 736–788 (in German); doi:10.1007/BF01206635, MR 1512302.Google Scholar
Iarrobino, A. and Kanev, V., Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721 (Springer, Berlin, 1999), appendix C by Iarrobino and Steven L. Kleiman; MR 1735271 (2001d:14056).CrossRefGoogle Scholar
Kahn, B., Démonstration géométrique du théorème de Lang-Néron et formules de Shioda–Tate, in Motives and algebraic cycles, Fields Institute Communications, vol. 56 (American Mathematical Society, Providence, RI, 2009), 149–155 (French, with English and French summaries); MR 2562456 (2010j:14083).Google Scholar
Katz, N. M., Sums of Betti numbers in arbitrary characteristic, Finite Fields Appl. 7 (2001), 29–44. Dedicated to Professor Chao Ko on the occasion of his 90th birthday; MR 1803934 (2002d:14028).Google Scholar
Keeler, D. S., Fujita’s conjecture and Frobenius amplitude, Amer. J. Math. 130 (2008), 1327–1336; doi:10.1353/ajm.0.0015, MR 2450210 (2009i:14006).Google Scholar
Kleiman, S. L., The Picard scheme, in Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123 (American Mathematical Society, Providence, RI, 2005), 235–321; MR 2223410.Google Scholar
Kloosterman, R., Elliptic K3 surfaces with geometric Mordell–Weil rank 15, Canad. Math. Bull. 50 (2007), 215–226; doi:10.4153/CMB-2007-023-2, MR 2317444 (2008f:14055).CrossRefGoogle Scholar
Lang, S., Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563; MR 0086367 (19,174a).Google Scholar
Liu, Q., Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford University Press, Oxford, 2002), translated from the French by Reinie Erné; MR 1917232 (2003g:14001).CrossRefGoogle Scholar
Madapusi Pera, K., The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. (2014), doi:10.1007/s00222-014-0557-5.Google Scholar
Madore, D. A. and Orgogozo, F., Calculabilité de la cohomologie étale modulo $\ell$, Preprint (2014), arXiv:1304.5376v3.Google Scholar
Mannoury, G., Surfaces-images, Nieuw Arch. Wisk. (2) 4 (1900), 112–129.Google Scholar
Maulik, D., Supersingular K3 surfaces for large primes. With an appendix by Andrew Snowden, Duke Math. J. 163 (2014), 2357–2425; doi:10.1215/00127094-2804783.CrossRefGoogle Scholar
Maulik, D. and Poonen, B., Néron–Severi groups under specialization, Duke Math. J. 161 (2012), 2167–2206; doi:10.1215/00127094-1699490, MR 2957700.Google Scholar
Milne, J. S., Étale cohomology, Princeton Mathematical Series, vol. 33 (Princeton University Press, Princeton, NJ, 1980); MR 559531 (81j:14002).Google Scholar
Minkowski, H., Zur Theorie der positiven quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196–202.Google Scholar
Néron, A., Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps, Bull. Soc. Math. France 80 (1952), 101–166 (in French);MR 0056951 (15,151a).Google Scholar
Nygaard, N. O., The Tate conjecture for ordinary K3 surfaces over finite fields, Invent. Math. 74 (1983), 213–237; doi:10.1007/BF01394314, MR 723215 (85h:14012).Google Scholar
Nygaard, N. and Ogus, A., Tate’s conjecture for K3 surfaces of finite height, Ann. of Math. (2) 122 (1985), 461–507; doi:10.2307/1971327, MR 819555 (87h:14014).Google Scholar
Oguiso, K., Shioda–Tate formula for an abelian fibered variety and applications, J. Korean Math. Soc. 46 (2009), 237–248; doi:10.4134/JKMS.2009.46.2.237, MR 2494474 (2009m:14011).Google Scholar
Deligne, P., Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie ($\mathit{SGA}~\mathit{4}{\textstyle \frac{1}{2}}$), Lecture Notes in Mathematics, vol. 569 (Springer, Berlin 1977); Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier; MR 0463174 (57 #3132).Google Scholar
Berthelot, P., Grothendieck, A. and Illusie, L., Théorie des intersections et théorème de Riemann–Roch, in Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Mathematics, vol. 225 (Springer, Berlin, 1971); Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre (French);MR 0354655 (50 #7133).Google Scholar
Shioda, T., On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59; MR 0429918 (55 #2927).Google Scholar
Shioda, T., An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986), 415–432; doi:10.2307/2374678, MR 833362 (87g:14033).CrossRefGoogle Scholar
Shioda, T., On the Mordell–Weil lattices, Comment. Math. Univ. St. Pauli 39 (1990), 211–240; MR 1081832 (91m:14056).Google Scholar
Simpson, C., Algebraic cycles from a computational point of view, Theoret. Comput. Sci. 392 (2008), 128–140; doi:10.1016/j.tcs.2007.10.008, MR 2394989 (2008m:14021).Google Scholar
Tate, J., Relations between K 2 and Galois cohomology, Invent. Math. 36 (1976), 257–274; MR 0429837 (55 #2847).CrossRefGoogle Scholar
Tate, J., Conjectures on algebraic cycles in l-adic cohomology, in Motives (Seattle, WA, 1991), Proceedings of Symposia in Applied Mathematics, vol. 55 (American Mathematical Society, Providence, RI, 1994), 71–83; MR 1265523 (95a:14010).Google Scholar
Tate, J., On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki 9 (1995), 415–440, Exp. No. 306; MR 1610977.Google Scholar
van Luijk, R., K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), 1–15; MR 2322921 (2008d:14058).Google Scholar
van Luijk, R., An elliptic K3 surface associated to Heron triangles, J. Number Theory 123 (2007), 92–119; doi:10.1016/j.jnt.2006.06.006, MR 2295433 (2007k:14077).Google Scholar
Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, vol. 2 (Springer, Berlin, 1998); with chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman; MR 1484973 (99c:13048).Google Scholar