Real multiplication through explicit correspondences

Abstract

We describe a method to compute equations for real multiplica- tion on the divisors of genus two curves via algebraic correspondences. We implement our method for various examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies{Kumar. We also compute a correspondence over the universal family over the Hilbert modular surface of discriminant 5 and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.