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Special points on products of modular curves. (English) Zbl 1072.14027

Author’s abstract: We prove the André-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let \(n\geq 0\), and let \(\Sigma\) be a subset of \({\mathbb C}^n\) consisting of points all of whose coordinates are \(j\)-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of \(\Sigma\) are {special sub-varieties}, i.e., determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant is proved unconditionally.

MSC:

14G35 Modular and Shimura varieties
14K22 Complex multiplication and abelian varieties
11G15 Complex multiplication and moduli of abelian varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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[1] Y. André, Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire , J. Reine Angew. Math. 505 (1998), 203–208. · Zbl 0918.14010
[2] F. Breuer, La conjecture d’André-Oort pour le produit de deux courbes modulaires de Drinfeld , C.R. Math. Acad. Sci. Paris 335 (2002), 867–870. · Zbl 1056.11036
[3] ——–, The André-Oort conjecture for products of Drinfeld modular curves , to appear in J. Reine Angew Math.
[4] L. Clozel and E. Ullmo, Équidistribution de sous-variétés spéciales , to appear in Ann. of Math (2). · Zbl 1099.11031
[5] P. B. Cohen and G. Wüstholz, “Application of the André-Oort conjecture to some questions in transcendence” in A Panorama of Number Theory, or the View from Baker’s Garden (Zürich, 1999) , ed. G. Wüstholz, Cambridge Univ. Press, Cambridge, 2002, 89–106. · Zbl 1051.11039
[6] C. Cornut, Non-trivialité des points de Heegner , C. R. Math. Acad. Sci. Paris 334 (2002), 1039–1042. · Zbl 1104.11030
[7] B. [S. J.] Edixhoven, Special points on the product of two modular curves , Compositio Math. 114 (1998), 315–328. · Zbl 0928.14019
[8] –. –. –. –., “On the André-Oort conjecture for Hilbert modular surfaces” in Moduli of Abelian Varieties , ed. C. Faber, G. van der Geer, and F. Oort, Progr. Math. 195 , Birkhäuser, Basel, 2001, 133–155. · Zbl 1051.68037
[9] ——–, four lectures on modular parametrisations and non-triviality of Heegner points, Summer School on the Birch and Swinnerton-Dyer Conjecture, Paris, July 2002, text at http://www.math.leidenuniv.nl/\(\tilde\;\)edix/public\(\_\)html\(\_\)rennes/talks/BSD2002.html, video at http://www.institut.math.jussieu.fr/BSD/video.html B. [S. J.] Edixhoven and A. Yafaev, Subvarieties of Shimura varieties , Ann. of Math. (2) 157 (2003), 621–645. JSTOR: · Zbl 1053.14023
[10] G. Faltings, Diophantine approximation on abelian varieties , Ann. of Math. (2) 133 (1991), 549–576. JSTOR: · Zbl 0734.14007
[11] R. Hartshorne, Algebraic Geometry , Grad. Texts in Math. 52 , Springer, New York, 1977. · Zbl 0367.14001
[12] A. J. de Jong, “Ample line bundles and intersection theory” in Diophantine Approximation and Abelian Varieties (Soesterberg, Netherlands, 1992) , ed. B. Edixhoven and J.-H. Evertse, Lecture Notes in Math. 1566 , Springer, Berlin, 1993, 69–76. · Zbl 0811.14005
[13] S. Lang, Algebra , 2nd ed., Addison-Wesley, Reading, Mass., 1984.
[14] ——–, Algebraic Number Theory , 2nd ed., Grad. Texts in Math. 110 , Springer, New York, 1994. · Zbl 0811.11001
[15] B. Moonen, Linearity properties of Shimura varieties, I , J. Algebraic Geom. 7 (1998), 539–567. · Zbl 0956.14016
[16] –. –. –. –., “Models of Shimura varieties in mixed characteristic” in Galois Representations in Arithmetic Algebraic Geometry (Durham, U.K., 1996) , ed. A. J. Scholl and R. L. Taylor, London Math. Soc. Lecture Note Ser. 254 , Cambridge Univ. Press, Cambridge, 1998, 267–350. · Zbl 0962.14017
[17] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev , Inst. Hautes Études Sci. Publ. Math. 54 (1981), 123–202. · Zbl 0496.12011
[18] J. H. Silverman, The Arithmetic of Elliptic Curves , Grad. Texts in Math. 106 , Springer, New York, 1986. · Zbl 0585.14026
[19] E. Ullmo, “Théorie ergodique et géométrie algébrique” in Proceedings of the International Congress of Mathematicians, Vol. 2 (Bejing, 2002) , Higher Ed. Press, Bejing, 2002, 197–206. · Zbl 1009.11046
[20] A. Yafaev, Special points on products of two Shimura curves , Manuscripta Math. 104 (2001), 163–171. · Zbl 0982.14016
[21] ——–, A conjecture of Yves André , · Zbl 1097.11032
[22] ——–, On a result of Moonen on the moduli space of principally polarized abelian varieties , to appear in Compositio Math. · Zbl 0811.14039
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