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The André–Oort conjecture for Drinfeld modular varieties

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  14 February 2013

Patrik Hubschmid
Patrik Hubschmid*
Affiliation:
Interdisciplinary Center for Scientific Computing, Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany (email: patrik.hubschmid@iwr.uni-heidelberg.de)
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Abstract

We consider the analogue of the André–Oort conjecture for Drinfeld modular varieties which was formulated by Breuer. We prove this analogue for special points with separable reflex field over the base field by adapting methods which were used by Klingler and Yafaev to prove the André–Oort conjecture under the generalized Riemann hypothesis in the classical case. Our result extends results of Breuer showing the correctness of the analogue for special points lying in a curve and for special points having a certain behaviour at a fixed set of primes.

Keywords

Drinfeld modular varietiesAndré–Oort conjecture

MSC classification

Secondary: 11G09: Drinfel'd modules; higher-dimensional motives, etc. 14G35: Modular and Shimura varieties

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 4 , April 2013 , pp. 507 - 567
Copyright
Copyright © 2013 The Author(s)

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References

[And98]André, Y., Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math. 505 (1998), 203–208.CrossRefGoogle Scholar
[Bor91]Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126, second edition (Springer, 1991).Google Scholar
[Bre05]Breuer, F., The André–Oort conjecture for products of Drinfeld modular curves, J. Reine Angew. Math. 579 (2005), 115–144.Google Scholar
[Bre07]Breuer, F., CM points on products of Drinfeld modular curves, Trans. Amer. Math. Soc. 359 (2007), 1351–1374.Google Scholar
[Bre12]Breuer, F., Special subvarieties of Drinfeld modular varieties, J. Reine Angew. Math. 668 (2012), 35–57.Google Scholar
[BP05]Breuer, F. and Pink, R., Monodromy groups associated to non-isotrivial Drinfeld modules in generic characteristic, in Number fields and function fields: two parallel worlds, Progress in Mathematics, vol. 239 (Birkhäuser, Boston, MA, 2005), 61–69.Google Scholar
[CU05]Clozel, L. and Ullmo, L., Equidistribution de sous-variétés spéciales, Ann. of Math. (2) 161 (2005), 1571–1588.Google Scholar
[Con99]Conrad, B., Irreducible components of rigid spaces, Ann. Inst. Fourier 49 (1999), 473–541.Google Scholar
[DH87]Deligne, P. and Husemöller, D., Survey of Drinfeld modules, in Current trends in arithmetical algebraic geometry, Arcata, CA, 1985, Contemporary Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 1987), 25–91.Google Scholar
[Dri74]Drinfeld, V. G., Elliptic modules, Mat. Sb. 94 (1974), 594–627, 656.Google Scholar
[Edi98]Edixhoven, B., Special points on the product of two modular curves, Compositio Math. 114 (1998), 315–328.Google Scholar
[Edi01]Edixhoven, B., On the André–Oort conjecture for Hilbert modular surfaces, in Moduli of abelian varieties, Texel, 1999, Progress in Mathematics, vol. 195 (Birkhäuser, Basel, 2001), 133–155.Google Scholar
[Edi05]Edixhoven, B., Special points on products of modular curves, Duke Math. J. 126 (2005), 325–348.Google Scholar
[EY03]Edixhoven, B. and Yafaev, A., Subvarieties of Shimura varieties, Ann. of Math. (2) 157 (2003), 621–645.Google Scholar
[FJ05]Fried, D. and Jarden, M., Field arithmetic, second edition (Springer, 2005).Google Scholar
[Ful98]Fulton, W., Intersection theory, second edition (Springer, 1998).CrossRefGoogle Scholar
[Gek89]Gekeler, E.-U., On the de Rham isomorphism for Drinfeld modules, J. Reine Angew. Math. 401 (1989), 188–208.Google Scholar
[Gek90]Gekeler, E.-U., De Rham cohomology and the Gauss–Manin connection for Drinfeld modules, in p-adic analysis, Trento, 1989, Lecture Notes in Mathematics, vol. 1454 (Springer, Berlin, 1990), 223–255.Google Scholar
[GW10]Görtz, U. and Wedhorn, T., Algebraic geometry 1: schemes, with examples and exercises (Vieweg+Teubner, 2010).Google Scholar
[Gos98]Goss, D., Basic structures of function field arithmetic (Springer, 1998).Google Scholar
[Gro64]Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Publ. Math. Inst. Hautes Études Sci. 20 (1964), 5–251.Google Scholar
[Gro66]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), 5–248.Google Scholar
[Har77]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, Berlin, 1977).Google Scholar
[Hei04]van der Heiden, G. J., Weil pairing for Drinfeld modules, Monatsh. Math. 143 (2004), 115–143.Google Scholar
[KY12]Klingler, B. and Yafaev, A., The André–Oort conjecture, Preprint (2012), arXiv:1209.0936 [math.NT].Google Scholar
[Koh11]Kohlhaase, J., Lubin–Tate and Drinfeld bundles, Tohoku Math. J. (2) 63 (2011), 217–254.Google Scholar
[Leh09]Lehmkuhl, T., Compactification of the Drinfeld modular surfaces, Mem. Amer. Math. Soc. 197 (2009), 921.Google Scholar
[Mum70]Mumford, D., Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5 (Oxford University Press, London, 1970).Google Scholar
[Neu07]Neukirch, J., Algebraische Zahlentheorie (Springer, 2007).Google Scholar
[Noo06]Noot, R., Correspondances de Hecke, action de Galois et la conjecture de André–Oort [d’après Edixhoven et Yafaev], Séminaire Bourbaki, Astérisque 307 (2006), 165–197, exp. no. 942.Google Scholar
[Pil11]Pila, J., $O$-minimality and the André–Oort conjecture for $\mathbb {C}^n$, Ann. of Math. (2) 173 (2011), 1779–1840.Google Scholar
[Pin97]Pink, R., The Mumford-Tate conjecture for Drinfeld-modules, Publ. Res. Inst. Math. Sci. 33 (1997), 393–425.Google Scholar
[Pin00]Pink, R., Strong approximation for Zariski dense subgroups over arbitrary global fields, Comment. Math. Helv. 75 (2000), 608–643.Google Scholar
[Pin12]Pink, R., Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank, Manuscripta Math., to appear; math.AG/1008.0013v4.Google Scholar
[Pra77]Prasad, G., Strong approximation for semi-simple groups over function fields, Ann. of Math. (2) 105 (1977), 553–572.Google Scholar
[Ser88]Serre, J-P., Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117 (Springer, Berlin, 1988).Google Scholar
[Sti93]Stichtenoth, H., Algebraic function fields and codes (Springer, 1993).Google Scholar
[UY12]Ullmo, E. and Yafaev, A., Galois orbits and equidistribution: towards the André–Oort conjecture, Preprint (2012), arXiv:1209.0934 [math.NT].Google Scholar
[Yaf06]Yafaev, A., A conjecture of Yves André’s, Duke Math. J. 132 (2006), 393–407.Google Scholar