Tuesday, January 14, 2020
01/14/2020 - 4:15pm
A Shimura-Teichmüller curve is a Teichmueller curve in the moduli space of Riemann surfaces that maps to a Shimura curve in the moduli space of Abelian varieties. The Shimura-Teichmüller curve property is also equivalent to having maximally many zero Lyapunov exponents in the Kontsevich-Zorich cocycle. Such curves are extremely rare and it was proven by Moeller that only one exists in genus 3, another exists in genus 4, and there are none in any other moduli space except the moduli space of genus 5 surfaces where he conjectured that none exist. We prove this conjecture using the solution to the jump problem of Hu and Norton to express higher order terms in the period matrix expansion near the boundary of moduli space and reduce the problem to a finite problem that can be solved by computer. This is joint work with Chaya Norton.