## Acta Mathematica

### The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

#### Abstract

In the moduli space $\mathcal{M}$g of genus-g Riemann surfaces, consider the locus $\mathcal{R}{\mathcal{M}_{\mathcal{O}}}$ of Riemann surfaces whose Jacobians have real multiplication by the order $\mathcal{O}$ in a totally real number field F of degree g. If g = 3, we compute the closure of $\mathcal{R}{\mathcal{M}_{\mathcal{O}}}$ in the Deligne–Mumford compactification of $\mathcal{M}$g and the closure of the locus of eigenforms over $\mathcal{R}{\mathcal{M}_{\mathcal{O}}}$ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $\mathcal{R}{\mathcal{M}_{\mathcal{O}}}$. Boundary strata of $\mathcal{R}{\mathcal{M}_{\mathcal{O}}}$ are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.

We apply this computation to give evidence for the conjecture that there are only finitely many algebraically primitive Teichmüller curves in $\mathcal{M}$3. In particular, we prove that there are only finitely many algebraically primitive Teichmüller curves generated by a 1-form having two zeros of order 3 and 1. We also present the results of a computer search for algebraically primitive Teichmüller curves generated by a 1-form having a single zero.

#### Article information

Source
Acta Math., Volume 208, Number 1 (2012), 1-92.

Dates
Revised: 17 March 2011
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892603

Digital Object Identifier
doi:10.1007/s11511-012-0074-6

Mathematical Reviews number (MathSciNet)
MR2910796

Zentralblatt MATH identifier
1250.14014

Rights

#### Citation

Bainbridge, Matt; Möller, Martin. The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3. Acta Math. 208 (2012), no. 1, 1--92. doi:10.1007/s11511-012-0074-6. https://projecteuclid.org/euclid.acta/1485892603

#### References

• A bikoff, W., Degenerating families of Riemann surfaces. Ann. of Math., 105 (1977), 29–44.
• A ccola, R., Differentials and extremal length on Riemann surfaces. Proc. Natl. Acad. Sci. USA, 46 (1960), 540–543.
• A hlfors, L., Lectures onQuasiconformal Mappings. Van Nostrand Mathematical Studies, 10. Van Nostrand, Toronto–New York–London, 1966.
• B ainbridge, M., Euler characteristics of Teichmüller curves in genus two. Geom. Topol., 11 (2007), 1887–2073.
• — Billiards in L-shaped tables with barriers. Geom. Funct. Anal., 20 (2010), 299–356.
• B ass, H., Torsion free and projective modules. Trans. Amer. Math. Soc., 102 (1962), 319–327.
• B elabas, K., A fast algorithm to compute cubic fields. Math. Comp., 66 (1997), 1213–1237.
• B ers, L., Spaces of degenerating Riemann surfaces, in Discontinuous Groupsand RiemannSurfaces (University of Maryland, College Park, MD, 1973), Ann. of Math. Studies, 79, pp. 43–55. Princeton Univ. Press, Princeton, NJ, 1974.
• — Finite-dimensional Teichmüller spaces and generalizations. Bull. Amer. Math. Soc., 5 (1981), 131–172.
• B ombieri, E., M asser, D. & Z annier, U., Intersecting a curve with algebraic subgroups of multiplicative groups. Int. Math. Res. Not., 1999 (1999), 1119–1140.
• B orel, A. & J i, L., Compactifications ofSymmetric andLocally SymmetricSpaces. Mathematics: Theory & Applications. Birkhäuser, Boston, MA, 2006.
• B orevich, A. & S hafarevich, I., Number Theory. Pure and Applied Mathematics, 20. Academic Press, New York, 1966.
• B ouw, I. & M öller, M., Teichmüller curves, triangle groups, and Lyapunov exponents. Ann. of Math., 172 (2010), 139–185.
• C alta, K., Veech surfaces and complete periodicity in genus two. J. Amer. Math. Soc., 17 (2004), 871–908.
• C ohen, H., Advanced Topicsin ComputationalNumber Theory. Graduate Texts in Mathematics, 193. Springer, New York, 2000.
• D ouady, A. & H ubbard, J., A proof of Thurston’s topological characterization of rational functions. Acta Math., 171 (1993), 263–297.
• E skin, A., M asur, H. & Z orich, A., Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants. Publ. Math. Inst. Hautes ÉtudesSci., 97 (2003), 61–179.
• F reitag, E., Hilbert ModularForms. Springer, Berlin-Heidelberg, 1990.
• van der G eer, G., Hilbert Modular Surfaces. Ergebnisse der Mathematik und ihrer Gren-zgebiete, 16. Springer, Berlin-Heidelberg, 1988.
• G oren, E., Lectures onHilbert ModularVarieties andModular Forms. CRM Monograph Series, 14. Amer. Math. Soc., Providence, RI, 2002.
• G riffiths, P. & H arris, J., Principles ofAlgebraic Geometry. Pure and Applied Mathematics. Wiley, New York, 1978.
• H abegger, P., Intersecting subvarieties of $\mathrm{G}_m^n$ with algebraic subgroups. Math. Ann., 342 (2008), 449–466.
• H arris, J. & M orrison, I., Moduli ofCurves. Graduate Texts in Mathematics, 187. Springer, New York, 1998.
• H indry, M. & S ilverman, J. H., Diophantine Geometry. Graduate Texts in Mathematics, 201. Springer, New York, 2000.
• van H oeij, M., An algorithm for computing the Weierstrass normal form of hyperelliptic curves. Preprint, 2002. arXiv:0203130 [math.AG].
• H ubert, P. & L anneau, E., Veech groups without parabolic elements. Duke Math. J., 133 (2006), 335–346.
• I mayoshi, Y. & T aniguchi, M., An Introductionto TeichmüllerSpaces. Springer, Tokyo, 1992.
• K enyon, R. & S millie, J., Billiards on rational-angled triangles. Comment. Math. Helv., 75 (2000), 65–108.
• L aurent, M., Équations diophantiennes exponentielles. Invent. Math., 78 (1984), 299–327.
• M asur, H., On a class of geodesics in Teichmüller space. Ann. of Math., 102 (1975), 205–221.
• — Extension of the Weil-Petersson metric to the boundary of Teichmüller space. Duke Math. J., 43 (1976), 623–635.
• M cmullen, C. T., Billiards and Teichmüller curves on Hilbert modular surfaces. J. Amer. Math. Soc., 16 (2003), 857–885.
• — Teichmüller curves in genus two: discriminant and spin. Math. Ann., 333 (2005), 87–130.
• — Prym varieties and Teichmüller curves. Duke Math. J., 133 (2006), 569–590.
• — Teichmüller curves in genus two: torsion divisors and ratios of sines. Invent. Math., 165 (2006), 651–672.
• — Dynamics of SL2(R) over moduli space in genus two. Ann. of Math., 165 (2007), 397–456.
• M öller, M., Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve. Invent. Math., 165 (2006), 633–649.
• — Variations of Hodge structures of a Teichmüller curve. J. Amer. Math. Soc., 19 (2006), 327–344.
• — Finiteness results for Teichmüller curves. Ann. Inst. Fourier (Grenoble), 58 (2008), 63–83.
• P hilippon, P., Sur des hauteurs alternatives. III. J. Math. Pures Appl., 74 (1995), 345–365.
• R oman, S., Advanced Linear Algebra. Graduate Texts in Mathematics, 135. Springer, New York, 1992.
• V eech, W., Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math., 97 (1989), 553–583.
• V oisin, C., Hodge Theory and Complex Algebraic Geometry. I. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge, 2007.
• W ard, C., Calculation of Fuchsian groups associated to billiards in a rational triangle. Ergodic Theory Dynam. Systems, 18 (1998), 1019–1042.
• W olpert, S., Riemann surfaces, moduli and hyperbolic geometry, in Lectures on Riemann Surfaces (Trieste, 1987), pp. 48–98. World Scientific, Teaneck, NJ, 1989.