Teichmüller curves in genus three and just likely intersections in G m n × G a n $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$

Abstract

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum M 3 (4), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmüller curves in terms of cross-ratios of six points on P 1 .These constraints are akin to those that appear in Zilber and Pink's conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of G n m × G n a (rather than the more standard G n m ). The ambient algebraic group lies outside the scope of Zilber's Conjecture but we are nonetheless able to prove a sufficiently strong height bound.For the generic stratum M 3 (1, 1, 1, 1), we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of G 9 m . These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.

The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve

“…Harder-Narasimhan filtration and degrees of eigenspace bundles. We order the line bundles L i in such a way that L 1 is the one that is generated by ω s (and is therefore maximal Higgs by [Möl06b]), and such that L 1 ⊕ L 2 is the next step in the Harder-Narasimhan filtration of V (see [BHM14,Prop. 4…”

“…The quotient of the degrees of the steps of the Harder-Narasimhan filtration and the degree of Ω 1 C (log(S) were computed in [YZ13, Table 1] for the (3, 1)-Stratum under the name w i of Weierstraß exponents by analyzing filtrations of the relative dualizing sheaf given by Weierstraß gap sequences. Using the same method, [BHM14,Prop. 4.3] shows that the Harder-Narasimhan filtration of V is given by the filtration of eigenspace bundles.…”

Lattice surfaces and smallest triangles

“…Split the surface in the direction of one of the saddle connections as in Figure 5, then the other saddle connection has to cross through either E 1 or E 2 . So, the length of their cross product has to be larger than min{c 1…”

“…McMullen [12] classified all the genus 2 lattice surfaces. Kenyon-Smillie [7], Bainbridge-Möller [2], Bainbridge-Habegger-Möller [1], Matheus-Wright [11], Lanneau-Nguyen-Wright [9] and others established finiteness results of Teichmüller curves in different settings. The forthcoming work of Eskin-Filip-Wright established finiteness result that are in a sense optimal.…”

Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci

“…Second, there are (in principle effective) finiteness results for algebraically primitive Teichmüller curves in the hyperelliptic components of ΩM g (g − 1, g − 1) ( [Möl08]) and in genus three ( [BHM16]). However, only for ΩM 2 (1, 1) there is a complete classification ( [McM06b]).…”