# Torsion on Jacobians and Teichmuller dynamics

In this post, I will describe a special case of a theorem about torsion points on Jacobians and Teichmuller dynamics which is proved in this paper. Namely, I will describe a different proof of a theorem of Moeller that for Teichmuller curves, zeroes of the holomorphic ${1}$-form must land in the torsion of (a factor of) the Jacobian. Some familiarity with Teichmuller dynamics will be assumed, as in the survey of Zorich.

Below the fold I will discuss some of the analytic ideas from variations of Hodge structures. Then I’ll try to connect these to Teichmuller dynamics and explain how this leads to various semisimplicity results. For applications, I’ll discuss how this implies that measurable ${{\mathrm{SL}}_2{\mathbb R}}$-invariant bundles have to be real-analytic. I’ll also mention how real multiplication on factors of the Jacobians arises.