On Shimura curves in the Schottky locus
Author:
Stefan Kukulies
Journal:
J. Algebraic Geom. 19 (2010), 371-397
DOI:
https://doi.org/10.1090/S1056-3911-09-00528-1
Published electronically:
August 18, 2009
MathSciNet review:
2580680
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Abstract |
References |
Additional Information
Abstract:
We show that a given rational Shimura curve $Y$ with strictly maximal Higgs field in the moduli space of $g$-dimensional principally polarized abelian varieties does not map to the closure of the Schottky locus for large $g$ if the generic point is the jacobian of a smooth curve.
We achieve this by using a result of Viehweg and Zuo which says that the corresponding family of abelian varieties over $Y$ is $Y$-isogenous to the $g$-fold product of a modular family of elliptic curves. After reducing the situation from the field of complex numbers to a finite field, we will see, combining the Weil and Sato-Tate conjectures, that for large $g$ no such family can become the jacobian of a family of curves.
References
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References
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- Igusa, J.: Fibre systems of Jacobian varieties. III. Fibre systems of elliptic curves, Amer. J. Math. 81, 1959, 453â476. MR 0104669 (21:3422)
- de Jong, Johan; Zhang, Shou-Wu: Generic Abelian Varieties with Real Multiplication are not Jacobians. Diophantine geometry, CRM Series, 4, 2007, 165â172. MR 2349653 (2009c:14085)
- Jost, J.; Zuo, K.: Arakelov type inequalities for Hodge bundles over algebraic varieties. I. Hodge bundles over algebraic curves. J. Algebraic Geom. 11 (2002), no. 3, 535â546. MR 1894937 (2003b:14016)
- Milne, J.S.: Ătale cohomology, Princeton Mathematical Series, 33, Princeton University Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
- Möller, M.; Viehweg, E.; Zuo, K.: Special families of curves, of abelian varieties, and of certain minimal manifolds over curves. Global aspects of complex geometry, 417â450, Springer, Berlin, 2006. MR 2264111 (2007k:14054)
- Möller, M.; Viehweg, E.; Zuo, K.: Stability of Hodge bundles and a numerical characterization of Shimura varieties. arXiv: 0706.3462, 2007.
- Möller, Martin: Shimura and TeichmĂŒller curves. arXiv: math.AG/0501333, 2005. MR 2150378 (2006e:14036)
- Oort, Frans; Steenbrink, Joseph: The local Torelli problem for algebraic curves. JournĂ©es de GĂ©ometrie AlgĂ©brique dâAngers, Juillet 1979, pp. 157â204. MR 605341 (82i:14014)
- Serre, Jean-Pierre: RĂ©partition asymptotique des valeurs propres de lâopĂ©rateur de Hecke $T_ p$. J. Amer. Math. Soc. 10 (1997), no. 1, 75â102. MR 1396897 (97h:11048)
- Silverman, Joseph H.: Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151. Springer-Verlag, New York, 1994. MR 1312368 (96b:11074)
- Szpiro, L.: Sur le thĂ©orĂšme de rigiditĂ© de Parsin et Arakelov. AstĂ©risque, 64, Soc. Math. France, Paris, 1979, pp. 169â202. MR 563470 (81f:14004)
- Tsfasman, M. A.; VlÄduĆŁ, S. G.: Asymptotic properties of zeta-functions. J. Math. Sci. (New York) 84 (1997), no. 5, 1445â1467. MR 1465522 (98h:11079)
- Viehweg, E.; Zuo, K.: Families over curves with a strictly maximal Higgs field. Asian J. Math. 7 (2003), no. 4, 575â598. MR 2074892 (2005j:14051)
- Viehweg, E.; Zuo, K.: A characterization of certain Shimura curves in the moduli stack of abelian varieties. J. Differential Geom. 66 (2004), no. 2, 233â287. MR 2106125 (2006a:14015)
- Viehweg, E.; Zuo, K.: Numerical bounds for semi-stable families of curves or of certain higher-dimensional manifolds. J. Algebraic Geom. 15 (2006), no. 4, 771â791. MR 2237270 (2007d:14019)
Additional Information
Stefan Kukulies
Affiliation:
UniversitÀt Duisburg-Essen, Mathematik, 45117 Essen, Germany
Email:
Stefan.Kukulies@uni-due.de
Received by editor(s):
March 20, 2008
Received by editor(s) in revised form:
December 26, 2008
Published electronically:
August 18, 2009
Additional Notes:
This work was financially supported by the Deutsche Forschungsgemeinschaft