Abstract
We compute the algebraic equation of the universal family over the Kenyon-Smillie $(2,3,4)$-Teichm\"uller curve and give a nice geometric description of the torsion map. Moreover, we re-prove independently that the found algebraic equation describes a Teichm\"uller curve by computing the Picard-Fuchs equation associated to the absolute cohomology bundle.
Highlights
Almost all known primitive Teichmuller curves fall in very few series
An infinite series in genus 2 is known by independent work of Calta and McMullen ([McM03], [Cal04]), which generalizes to the construction of the infinite
The universal family over the complement of the orbifold point of the Kenyon-Smillie (2, 3, 4)-Teichmuller curve is given by the family of plane quartics satisfying the equation
Summary
Almost all known primitive Teichmuller curves fall in very few series. Currently, an infinite series in genus 2 is known by independent work of Calta and McMullen ([McM03], [Cal04]), which generalizes to the construction of the infinite. The universal family over the complement of the orbifold point of the Kenyon-Smillie (2, 3, 4)-Teichmuller curve is given by the family of plane quartics satisfying the equation (1.1). The initial motivation of the paper was to investigate a question of Alex Wright about the relation between real multiplication and the torsion map One can see such a relation in the Veech-Ward-Bouw-Moller Teichmuller curves, where real multiplication is induced by the correspondence given by the graph of an automorphism coming from the Galois normalization map (see Section 4.7 for details). As in the Veech-Ward-Bouw-Moller case, the Kenyon-Smillie Teichmuller curve is uniformized by a triangle group. We will deduce from Proposition 1.2 that for the Kenyon-Smillie-Teichmuller curve, real multiplication does not come from the normalization of the torsion map. A related open question is whether real multiplication for this curve is of Hecke type as in the Veech-Ward-Bouw-Moller case (cf. [Wri13])
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