Available in electronic format
Available in print format		 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN: 1088-6834(e) ISSN: 0894-0347(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Variations of Hodge structures of a Teichmüller curve

Author(s): Martin Möller
Journal: J. Amer. Math. Soc. 19 (2006), 327-344.
MSC (2000): Primary 32G15; Secondary 14D07
Posted: December 12, 2005
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Teichmüller curves are geodesic discs in Teichmüller space that project to an algebraic curve in the moduli space $ M_g$. We show that for all $ g \geq 2$ Teichmüller curves map to the locus of real multiplication in the moduli space of abelian varieties. Observe that McMullen has shown that precisely for $ g=2$ the locus of real multiplication is stable under the $ {\rm SL}_2({\mathbb{R}})$-action on the tautological bundle $ \Omega M_g$.

We also show that Teichmüller curves are defined over number fields and we provide a completely algebraic description of Teichmüller curves in terms of Higgs bundles. As a consequence we show that the absolute Galois group acts on the set of Teichmüller curves.

References:

[CoWo90]
Cohen, P., Wolfart, J., Modular embeddings for some non-arithmetic Fuchsian groups, Acta Arithm. 56 (1990), 93-110. MR 1075639 (92d:11039)

[De70]
Deligne, P., Equations Différentielles à Points Singuliers Réguliers, Springer LNM 163 (1970). MR 0417174 (54:5232)

[De71]
Deligne, P., Théorie de Hodge II, Publ. Math. I.H.E.S. 40 (1971), 5-57. MR 0498551 (58:16653a)

[De87]
Deligne, P., Un théorème de finitude pour la monodromie, Discrete groups in Geometry and Analysis, Birkhäuser, Progress in Math. 67 (1987), 1-19. MR 0900821 (88h:14013)

[Fa83]
Faltings, G., Arakelov's Theorem for Abelian varieties, Invent. Math. 73 (1983), 337-347. MR 0718934 (85m:14061)

[FuHa91]
Fulton, W., Harris, J., Representation theory, GTM 129, Springer (1991). MR 1153249 (93a:20069)

[GuJu00]
Gutkin, E., Judge, C., Affine mappings of translation surfaces, Duke Math. J. 103 No. 2 (2000), 191-212.MR 1760625 (2001h:37071)

[Ko85]
Kollár, J., Subadditivity of the Kodaira dimension: Fibers of general type, Adv. Studies in Pure Math. 10 (1987), 361-398. MR 0946244 (89i:14029)

[Lo03]
Lochak, P., On arithmetic curves in the moduli space of curves, Journal of the Math. Inst. of Jussieu 4, No. 3 (2005), 443-508.

[LaBi92]
Lange, H., Birkenhake, C., Complex abelian varieties, Springer Grundlehren 302 (1992). MR 1217487 (94j:14001)

[McM1]
McMullen, C., Billiards and Teichmüller curves on Hilbert modular surfaces, Journal of the AMS 16 (2003), 857-885. MR 1992827 (2004f:32015)

[McM2]
McMullen, C., Teichmüller geodesics of infinite complexity, Acta Math. 191, No. 2 (2003), 191-223. MR 2051398 (2005e:32025)

[McM3]
McMullen, C., Dynamics of $ {\rm SL}_2({\mathbb{R}})$ over moduli space in genus two, preprint (2003).

[Mo98]
Moonen, B., Linearity properties of Shimura varieties I, J. Algebraic Geom. 7 (1998), 539-567.MR 1618140 (99e:14028)

[Mo03]
Möller, M., Teichmüller curves, Galois actions, $ \widehat{GT}$-relations, Math. Nachr. 278, No. 9 (2005), 1061-1077.MR 2150378

[Mu66]
Mumford, D., Families of Abelian varieties, Proc. Symp. Pure Math. 9 (1966), 347-351. MR 0206003 (34:5828)

[Mu69]
Mumford, D., A note of Shimura's paper: Discontinuous groups and Abelian varieties, Math. Ann. 181 (1969), 345-351. MR 0248146 (40:1400)

[Sc96]
Schoen, C., Varieties dominated by product varieties, Int. J. Math. 7 (1996), 541-571. MR 1408839 (97f:14050)

[Sd73]
Schmid, W., Variation of Hodge Structure: The singularities of the period mapping, Inv. Math. 22 (1973), 211-319.MR 0382272 (52:3157)

[ScWo00]
Schmutz-Schaller, P., Wolfart, J., Semi-arithmetic Fuchsian groups and modular embeddings, J. London Math. Soc. 61 (2000), 13-24. MR 1745404 (2001a:11071)

[Si90]
Simpson, C., Harmonic bundles on noncompact curves, Journal of the AMS 3 (1990), 713-769. MR 1040197 (91h:58029)

[Sh66]
Shimura, G., Moduli of Abelian Varieties and Number Theory, Proc. Symp. Pure Math. 9 (1966), 313-332. MR 0237507 (38:5789)

[SmWe04]
Smillie, J., Weiss, B., Minimal sets for flows on moduli space, Israel J. of Math. 142 (2004), 249-260. MR 2085718 (2005g:37067)

[Ta69]
Takeuchi, K., On some discrete subgroups of $ {\rm SL}_2({\mathbb{R}})$, J. Fac. Sci. Univ. Tokyo 16 (1969), 97-100. MR 0262171 (41:6781)

[Ve89]
Veech, W., Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), 533-583. MR 1005006 (91h:58083a)

[vG87]
van der Geer, G., Hilbert modular surfaces, Erg. der Math., Bd. 16, Springer-Verlag (1987). MR 0930101 (89c:11073)

[ViZu04]
Viehweg, E., Zuo, K., A characterization of Shimura curves in the moduli stack of abelian varieties, Journal of Diff. Geometry 66 (2004), 233-287.MR 2106125

[ViZu05]
Viehweg, E., Zuo, K., Numerical bounds for semi-stable families of curves or of higher-dimensional manifolds, preprint (2005).

Similar Articles:

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 32G15, 14D07

Retrieve articles in all Journals with MSC (2000): 32G15, 14D07

Additional Information:

Martin Möller
Affiliation: Universität Essen, FB 6 (Mathematik), 45117 Essen, Germany
Email: martin.moeller@uni-essen.de

DOI: 10.1090/S0894-0347-05-00512-6
PII: S 0894-0347(05)00512-6
Keywords: Teichm\"uller curve, real multiplication, maximal Higgs local subsystem
Received by editor(s): January 26, 2004
Posted: December 12, 2005
Copyright of article: Copyright 2005, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google