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

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \Omega\mathcal{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 \mathbf{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 \mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n} (rather than the more standard \mathbf{G}_{m}^{n}). 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 \Omega\mathcal{M}_{3}(1,1,1,1), we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of \mathbf{G}_{m}^{9}. 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.

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