English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We prove that the moduli space of compact genus three Riemann surfacescontains only finitely many algebraically primitive Teichmueller curves. Forthe stratum consisting of holomorphic one-forms in genus three with a singlezero, our approach to finiteness uses the Harder-Narasimhan filtration of theHodge bundle over a Teichmueller curve to obtain new information on thelocations of the zeros of eigenforms. By passing to the boundary of modulispace, this gives explicit constraints on the cusps of Teichmueller curves interms of cross-ratios of six points on a projective line. These constraints areakin to those that appear in Zilber and Pink's conjectures on unlikelyintersections in diophantine geometry. However, in our case one is leadnaturally to the intersection of a surface with a family of codimension twoalgebraic subgroups of $G_m^n \times G_a^n$ (rather than the more standard$G_m^n$). The ambient algebraic group lies outside the scope of Zilber'sConjecture but we are nonetheless able to prove a sufficiently strong heightbound. For the generic stratum in genus three, we obtain global torsion orderbounds through a computer search for subtori of a codimension-two subvariety of$G_m^9$. These torsion bounds together with new bounds for the moduli ofhorizontal cylinders in terms of torsion orders yields finiteness in thisstratum. The intermediate strata are handled with a mix of these techniques.

Teichmueller curves in genus three and just likely intersections in $G_m^n x G_a^n$