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Real multiplication through explicit correspondences

Part of: Deformations of analytic structures Curves Algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  26 August 2016

Abhinav Kumar  and
Ronen E. Mukamel
Abhinav Kumar
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA email thenav@gmail.com
Ronen E. Mukamel
Affiliation:
Department of Mathematics, Rice University MS 136, 6100 Main St., Houston, TX 77005, USA email ronen@rice.edu

Abstract

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We compute equations for real multiplication on the divisor classes of genus-2 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.

MSC classification

Primary: 14-04: Explicit machine computation and programs (not the theory of computation or programming) 14H40: Jacobians, Prym varieties
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory 14G35: Modular and Shimura varieties

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