Real multiplication through explicit correspondences

TL;DR

This paper develops explicit algebraic correspondences to compute real multiplication on genus two curves, using van Wamelen's method and modular surface models, and applies these to prove a conjecture in Riemann surface dynamics.

Contribution

It introduces a practical method for computing real multiplication on Jacobians of genus two curves via explicit correspondences and applies it to a conjecture in moduli space dynamics.

Findings

Computed equations for real multiplication on genus two curves.

Established a correspondence over the universal family for discriminant 5.

Proved a conjecture of A. Wright on Riemann surface dynamics.

Abstract

We compute equations for real multiplication on the divisor classes of genus two curves via algebraic correspondences. We do so by implementing van Wamelen's method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for 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.