Real multiplication through explicit correspondences

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.

15 pages, to be presented at ANTS XII

Cited by in corpus (8)

• Rigorous computation of the endomorphism ring of a Jacobian
• Marked points on translation surfaces
• Computing the geometric endomorphism ring of a genus 2 Jacobian
• GL(2,R)-Invariant Measures in Marked Strata: Generic Marked Points, Earle-Kra for Strata, and Illumination
• Periodic Points in Genus Two: Holomorphic Sections over Hilbert Modular Varieties, Teichmuller Dynamics, and Billiards
• Moduli for rational genus 2 curves with real multiplication for discriminant 5
• Existence of closed geodesics through a regular point on translation surfaces
• Diagonals of rational functions: from differential algebra to effective algebraic geometry (unabridged version)