Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-28T08:58:19.200Z Has data issue: false hasContentIssue false

On the Oort conjecture for Shimura varieties of unitary and orthogonal types

Published online by Cambridge University Press:  02 February 2016

Ke Chen
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China School of Mathematics, University of Science and Technology of China, Hefei 230026, China email kechen@ustc.edu.cn
Xin Lu
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200241, China email luxin001@uni-mainz.de Current address: Institut für Mathematik, Universität Mainz, 55099 Mainz, Germany
Kang Zuo
Affiliation:
Institut für Mathematik, Universität Mainz, 55099 Mainz, Germany email zuok@uni-mainz.de
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.

In this paper we study the Oort conjecture concerning the non-existence of Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety ${\mathcal{A}}_{g}$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibered surfaces, we show that a Shimura curve $C$ is not contained generically in the Torelli locus if its canonical Higgs bundle contains a unitary Higgs subbundle of rank at least $(4g+2)/5$. From this we prove that a Shimura subvariety of $\mathbf{SU}(n,1)$ type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus $g$, the dimension $n+1$, the degree $2d$ of CM field of the Hermitian space, and the type of the symplectic representation defining the Shimura subdatum. A similar result holds for Shimura subvarieties of $\mathbf{SO}(n,2)$ type, defined by spin groups associated to quadratic spaces over a totally real number field of degree at least $6$ subject to some natural constraints of signatures.

Type
Research Article
Copyright
© The Authors 2016 

References

André, Y., Mumford-Tate groups and theorem of the fixed part, Compositio Math. 82 (1992), 124.Google Scholar
Baily, W. and Borel, A., Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442528.Google Scholar
Coleman, R. F., Torsion points on curves, in Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986), Advanced Studies in Pure Mathematics, vol. 129 (North-Holland, Amsterdam, 1987), 235247.Google Scholar
Deligne, P., Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, OR, 1977), Part 2, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 247289.Google Scholar
de Jong, J. and Noot, R., Jacobians with complex multiplication, in Arithmetic algebraic geometry (Texel, 1989), Progress in Mathematics, vol. 89 (Birkhäuser, Boston, 1991), 177192.Google Scholar
de Jong, J. and Zhang, S.-W., Generic Abelian varieties with real multiplication are not Jacobians, in Diophantine Geometry, CRM Series, vol. 4 (Edizioni della Normale, Pisa, 2007), 165172.Google Scholar
Dwork, B. and Ogus, A., Canonical liftings of Jacobians, Compositio Math. 58 (1986), 111131.Google Scholar
Frediani, P., Ghigi, A. and Penegini, M., Shimura varieties in the Torelli locus via Galois coverings, Preprint (2014), arXiv:1402.0973v2.CrossRefGoogle Scholar
Fujita, T., On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), 779794.Google Scholar
Goodman, R. and Wallach, N., Symmetry, invariants, and representations, Graduate Texts in Mathematics, vol. 255 (Springer, Dordrecht, 2009).Google Scholar
Hain, R., Locally symmetric families of curves and Jacobians, in Moduli of curves and Abelian varieties, Aspects of Mathematics, vol. E33 (Vieweg, Braunschweig, 1999), 91108.CrossRefGoogle Scholar
Harder, G. and Narasimhan, M. S., On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215248.Google Scholar
Hida, H., p-adic automorphic forms on Shimura varieties (Springer, New York, 2004).Google Scholar
Humphreys, J., Linear algebraic groups, Graduate Texts in Mathematics, vol. 21 (Springer, New York, 1981).Google Scholar
Kudla, S., Algebraic cycles on Shimura varieties of orthogonal type, Duke J. Math. 86 (1997), 3978.CrossRefGoogle Scholar
Kudla, S. and Rapoport, M., Special cycles on unitary Shimura varieties II: global theory, J. Reine Angew. Math. 697 (2014), 91157.Google Scholar
Kukulies, S., On Shimura curves in the Schottky locus, J. Algebraic Geom. 19 (2010), 371397.Google Scholar
Lu, X. and Zuo, K., On the slope of hyperelliptic fibrations with positive relative irregularity, Trans. Amer. Math. Soc., to appear, arXiv:1311.7271.Google Scholar
Lu, X. and Zuo, K., The Oort conjecture on Shimura curves in the Torelli locus of curves, Preprint (2014), arXiv:1405.4751.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), 265378.Google Scholar
Mohajer, A. and Zuo, K., On Shimura subvarieties generated by families of abelian covers of $\mathbb{P}^{1}$, Preprint (2014), arXiv:1402.1900.Google Scholar
Moonen, B., Linearity properties of Shimura varieties, I, J. Algebraic Geom. 7 (1998), 539567.Google Scholar
Moonen, B., Special subvarieties arising from families of cyclic covers of the projective line, Doc. Math. 15 (2010), 793819.Google Scholar
Moonen, B. and Oort, F., The Torelli locus and special subvarieties, in Handbook of moduli. Vol. II, Advanced Lectures in Mathematics, vol. 25 (International Press, Somerville, MA, 2013), 549594.Google Scholar
Mumford, D., Geometric invariant theory, in Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34 (Springer, New York, 1965).Google Scholar
Noot, R., Correspondances de Hecke, action de Galois et la conjecture d’André–Oort (d’après Edixhoven et Yafaev), in Séminaire Bourbaki: Volume 2004/2005, Astérisque, vol. 307 (Société Mathématique de France, 2006), vii, 165197, Exposé 942.Google Scholar
Oort, F., Canonical liftings and dense sets of CM-points, in Arithmetic geometry (Cortona, 1994), Symposia Mathematica, vol. XXXVII (Cambridge University Press, Cambridge, 1997), 228234.Google Scholar
Satake, I., Holomorphic imbeddings of symmetric domains into a Siegel space, Amer. J. Math. 87 (1965), 425461.Google Scholar
Satake, I., Symplectic representations of algebraic groups, in Algebraic groups and discontinuous subgroups, Proceedings of Symposia in Pure Mathematics 9 (American Mathematical Society, Providence, RI, 1966), 352357.CrossRefGoogle Scholar
Satake, I., Symplectic representations of algebraic groups satisfying a certain analyticity condition, Acta Math. 117 (1967), 215279.Google Scholar
Scanlon, T., A proof of the André–Oort conjecture via mathematical logic, after Pila, Wilkie, and Zannier, in Séminaire Bourbaki: Volume 2010–2011, Astérisque, vol. 348 (Société Mathématique de France, 2012), ix, 299315, Exposé 1037.Google Scholar
Viehweg, E. and Zuo, K., A characterization of certain Shimura curves in the moduli stack of abelian varieties, J. Differential Geom. 66 (2004), 233287.Google Scholar
Xiao, G., Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), 449466.Google Scholar
Xiao, G., Irregular families of hyperelliptic curves, in Algebraic geometry and algebraic number theory (Tianjin, 1989–1990), Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 3 (World Scientific, River Edge, NJ, 1992), 152156.Google Scholar