Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-31T03:08:50.782Z Has data issue: false hasContentIssue false
  • English
  • Français

Finiteness theorems for K3 surfaces and abelian varieties of CM type

Part of: Surfaces and higher-dimensional varieties Arithmetic algebraic geometry (Co)homology theory

Published online by Cambridge University Press:  18 July 2018

Martin Orr  and
Alexei N. Skorobogatov
Martin Orr
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2BZ, UK email m.orr@imperial.ac.uk
Alexei N. Skorobogatov
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2BZ, UK email a.skorobogatov@imperial.ac.uk Institute for the Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow 127994, Russia
Get access Rights & Permissions [Opens in a new window]

Abstract

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

Keywords

abelian varietiesK3 surfacescomplex multiplicationBrauer group

MSC classification

Primary: 11G15: Complex multiplication and moduli of abelian varieties 14J28: $K3$ surfaces and Enriques surfaces 14F22: Brauer groups of schemes

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 8 , August 2018 , pp. 1571 - 1592
Copyright
© The Authors 2018 

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.)

Article purchase

Temporarily unavailable

References

André, Y., On the Shafarevich and Tate conjectures for hyperkähler varieties , Math. Ann. 305 (1996), 205–248.Google Scholar
Andreatta, F., Goren, E. Z., Howard, B. and Madapusi Pera, K., Faltings heights of abelian varieties with complex multiplication , Ann. of Math. (2) 187 (2018), 391–531.Google Scholar
Baily, W. L. Jr. and Borel, A., Compactification of arithmetic quotients of bounded symmetric domains , Ann. of Math. (2) 84 (1966), 442–528.Google Scholar
Bogomolov, F. A., Points of finite order on abelian varieties (Russian) , Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 782–804.Google Scholar
Borovoi, M. V., Langlands’ conjecture concerning conjugation of connected Shimura varieties , Selecta Math. Soviet. 3 (1983/84), 3–39.Google Scholar
Bridgeland, T. and Maciocia, A., Complex surfaces with equivalent derived categories , Math. Z. 236 (2001), 677–697.Google Scholar
Bruin, N., Flynn, V. E., González, J. and Rotger, V., On finiteness conjectures for endomorphism algebras of abelian surfaces , Math. Proc. Cambridge Philos. Soc. 141 (2006), 383–408.Google Scholar
Cadoret, A. and Kret, A., Galois-generic points on Shimura varieties , Algebra Number Theory 10 (2016), 1893–1934.Google Scholar
Cadoret, A. and Moonen, B., Integral and adelic aspects of the Mumford–Tate conjecture, Preprint (2015), arXiv:1508.06426.Google Scholar
Cassels, J. W. S., Rational quadratic forms, London Mathematical Society Monographs, vol. 13 (Academic Press, London, 1978).Google Scholar
Charles, F., Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for K3 surfaces, and the Tate conjecture , Ann. of Math. (2) 184 (2016), 487–526.Google Scholar
Colliot-Thélène, J.-L. and Skorobogatov, A. N., Descente galoisienne sur le groupe de Brauer , J. Reine Angew. Math. 682 (2013), 141–165.Google Scholar
Deligne, P., Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques , in Automorphic forms, representations and L-functions (Part 2), Proceedings of Symposia in Pure Mathematics, XXXIII (American Mathematical Society, Providence, RI, 1979), 247–289.Google Scholar
Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D., Analytic pro-p groups, London Mathematical Society Lecture Note Series, vol. 157 (Cambridge University Press, 1991).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III , Publ. Math. Inst. Hautes Études Sci. 28 (1966).Google Scholar
Grothendieck, A., Le groupe de Brauer I, II, III , in Dix exposés sur la cohomologie des schémas (North-Holland, 1968), 46–188.Google Scholar
Henniart, G., Représentations l-adiques abéliennes , in Séminaire de Théorie des Nombres, Paris 1980–81, Progress in Mathematics, vol. 22 (Birkhäuser, Boston, MA, 1982), 107–126.Google Scholar
Huybrechts, D., Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, vol. 158 (Cambridge University Press, 2016).Google Scholar
Ieronymou, E., Diagonal quartic surfaces and transcendental elements of the Brauer groups , J. Inst. Math. Jussieu 9 (2010), 769–798.Google Scholar
Ieronymou, E. and Skorobogatov, A. N., Odd order Brauer–Manin obstruction on diagonal quartic surfaces , Adv. Math. 270 (2015), 181–205; Corrigendum: Adv. Math. 307 (2017), 1372–1377.Google Scholar
Ieronymou, E., Skorobogatov, A. N. and Zarhin, Yu. G., On the Brauer group of diagonal quartic surfaces , J. Lond. Math. Soc. (2) 83 (2011), 659–672.Google Scholar
Larsen, M. and Pink, R., A connectedness criterion for l-adic Galois representations , Israel J. Math. 97 (1997), 1–10.Google Scholar
Madapusi Pera, K., The Tate conjecture for K3 surfaces in odd characteristic , Invent. Math. 201 (2015), 625–668.Google Scholar
Margulis, G. A., Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 17 (Springer, Berlin, 1991).Google Scholar
Masser, D. and Wüstholz, G., Isogeny estimates for abelian varieties, and finiteness theorems , Ann. of Math. (2) 137 (1993), 459–472.Google Scholar
Milne, J. S., Complex multiplication (version 0.00). Online notes, available at jmilne.org/math/CourseNotes/cm.html.Google Scholar
Milne, J. S., The action of an automorphism of C on a Shimura variety and its special points , in Arithmetic and geometry, Vol. I, Progress in Mathematics, vol. 35 (Birkhäuser Boston, Boston, MA, 1983), 239–265.Google Scholar
Milne, J. S., Abelian varieties , in Arithmetic geometry (Storrs, CT, 1984) (Springer, New York, 1986), 103–150.Google Scholar
Milne, J. S., Introduction to Shimura varieties , in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 265–378.Google Scholar
Minkowski, H., Zur Theorie der positiven quadratischen Formen , J. Reine Angew. Math. 101 (1887), 196–202.Google Scholar
Newton, R., Transcendental Brauer groups of products of CM elliptic curves , J. Lond. Math. Soc. (2) 93 (2016), 397–419.Google Scholar
Nikulin, V. V., Integer symmetric bilinear forms and some of their geometric applications , Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177.Google Scholar
Nori, M. V., On subgroups of GL n (F p ) , Invent. Math. 88 (1987), 257–275.Google Scholar
Orlov, D. O., Equivalences of derived categories and K3 surfaces , J. Math. Sci. (N.Y.) 84 (1997), 1361–1381.Google Scholar
Piatetski-Shapiro, I. I. and Shafarevich, I. R., Torelli’s theorem for algebraic surfaces of type K3 , Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572.Google Scholar
Piatetski-Shapiro, I. I. and Shafarevich, I. R., The arithmetic of surfaces of type K3, in Proceedings of the international conference on number theory (Moscow, 1971), Trudy Matematicheskogo Instituta Imeni V. A. Steklova 132 (1973), 44–54.Google Scholar
Pila, J. and Tsimerman, J., The André–Oort conjecture for the moduli space of abelian surfaces , Compositio Math. 149 (2013), 204–216.Google Scholar
Pila, J. and Tsimerman, J., Ax–Lindemann for A g , Ann. of Math. (2) 179 (2014), 659–681.Google Scholar
Pink, R., Arithmetical compactification of mixed Shimura varieties, Bonner Mathematische Schriften, vol. 209 (Universität Bonn, Mathematisches Institut, Bonn, 1990).Google Scholar
Pink, R., A combination of the conjectures of Mordell–Lang and André–Oort , in Geometric methods in algebra and number theory, Progress in Mathematics, vol. 235 (Birkhäuser Boston, Boston, MA, 2005), 251–282.Google Scholar
Pohlmann, H., Algebraic cycles on abelian varieties of complex multiplication type , Ann. of Math. (2) 88 (1968), 161–180.Google Scholar
Rizov, J., Kuga–Satake abelian varieties of K3 surfaces in mixed characteristic , J. Reine Angew. Math. 648 (2010), 13–67.Google Scholar
Serre, J.-P., Cours d’arithmétique (Presses Universitaires de France, Paris, 1970).Google Scholar
Serre, J.-P., Représentations ℓ-adiques , in Algebraic number theory (Kyoto Internat. Sympos., RIMS, Univ. Kyoto, 1976) (Japan Soc. Promotion Sci., Tokyo, 1977), 177–193.Google Scholar
Serre, J.-P., Propriétés conjecturales des groupes de Galois motiviques et des représentations ℓ-adiques , in Motives (Seattle, WA, 1991), Proceedings of Symposia in Pure Mathematics, vol. 55 (American Mathematical Society, Providence, RI, 1994), 377–400; part 1.Google Scholar
Serre, J.-P., Lettre à Ken Ribet, 1/1/1981, Œuvres IV (Springer, Berlin, 2000), 1–17.Google Scholar
Shafarevich, I. R., On the arithmetic of singular K3-surfaces , in Algebra and analysis (Kazan, 1994) (De Gruyter, Berlin, 1996), 103–108.Google Scholar
She, Y., The unpolarized Shafarevich conjecture for K3 surfaces, Preprint (2017),arXiv:1705.09038.Google Scholar
Skorobogatov, A. N. and Zarhin, Yu. G., A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces , J. Algebraic Geom. 17 (2008), 481–502.Google Scholar
Taelman, L., K3 surfaces over finite fields with given L-function , Algebra Number Theory 10 (2016), 1133–1146.Google Scholar
Tankeev, S. G., Surfaces of type K3 over number fields and the Mumford–Tate conjecture , Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), 179–206.Google Scholar
Tsimerman, J., The André–Oort conjecture for A g , Ann. of Math. (2) 187 (2018), 379–390.Google Scholar
Várilly-Alvarado, A., Arithmetic of K3 surfaces , in Geometry over nonclosed fields, Simons Symposia, vol. 5, eds Bogomolov, F., Hassett, B. and Tschinkel, Y. (Springer, 2017), 197–248.Google Scholar
Várilly-Alvarado, A. and Viray, B., Abelian n-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces , Forum Math. Sigma 5 (2017), e26.Google Scholar
Vasiu, A., Some cases of the Mumford–Tate conjecture and Shimura varieties , Indiana Univ. Math. J. 57 (2008), 1–75.Google Scholar
Weil, A., Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144 (Springer, New York, 1967).Google Scholar
Yuan, X. and Zhang, S., On the averaged Colmez conjecture , Ann. of Math. (2) 187 (2018), 533–638.Google Scholar
Zarhin, Yu. G., Hodge groups of K3 surfaces , J. Reine Angew. Math. 341 (1983), 193–220.Google Scholar
Zarhin, Yu. G., A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction , Invent. Math. 79 (1985), 309–321.Google Scholar