Abstract
Teichmuller curves are geodesic discs in Teichmuller space that project to an algebraic curve in the moduli space Mg . We show that for all g ≥ 2 Teichmuller 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 SL2(R)-action on the tautological bundle ΩMg . We also show that Teichmuller curves are defined over number fields and we provide a completely algebraic description of Teichmuller curves in terms of Higgs bundles. As a consequence we show that the absolute Galois group acts on the set of Teichmuller curves.
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