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Real multiplication through explicit correspondences

Part of: Algebraic geometry Deformations of analytic structures Curves 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 [email protected]
Ronen E. Mukamel
Affiliation:
Department of Mathematics, Rice UniversityMS 136, 6100 Main St., Houston, TX 77005, USA email [email protected]

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
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 29 - 42
Copyright
© The Author(s) 2016 

References

Birkenhake, C. and Lange, H., Complex abelian varieties , 2nd edn, Grundlehren der mathematischen Wissenschaften 302 (Springer, Berlin, 2004).Google Scholar
Bosma, W., Cannon, J. J., Fieker, C. and Steel, A. (eds), Handbook of Magma functions , available online at http://magma.maths.usyd.edu.au/magma/handbook/.Google Scholar
Cassels, J. W. S. and Flynn, E. V., Prolegomena to a middlebrow arithmetic of curves of genus 2 , London Mathematical Society Lecture Note Series 320 (Cambridge University Press, Cambridge, 1996).Google Scholar
Elkies, N. D. and Kumar, A., ‘K3 surfaces and equations for Hilbert modular surfaces’, Algebra Number Theory 8 (2014) no. 10, 2297–2411.Google Scholar
Eskin, A., McMullen, C. T., Mukamel, R. E. and Wright, A., ‘Billiards in quadrilaterals, Hurwitz spaces, and real multiplication of Hecke type’, in preparation.Google Scholar
Filip, S., ‘Splitting mixed Hodge structures over affine invariant manifolds’, Ann. of Math. (2) 183 (2016) 681–713.Google Scholar
Hashimoto, K. and Sakai, Y., ‘On a versal family of curves of genus two with √2-multiplication’, Algebraic number theory and related topics 2007 , RIMS Kôkyûroku Bessatsu B12 (Res. Inst. Math. Sci. (RIMS), Kyoto, 2009) 249–261.Google Scholar
Hashimoto, K. and Sakai, Y., ‘General form of Humbert’s modular equation for curves with real multiplication of 𝛥 = 5’, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009) no. 10, 171–176.Google Scholar
Humbert, G., ‘Sur les fonctionnes abéliennes singulières’, J. Math. Pures Appl. série 5 t. V (1899) 233–350.Google Scholar
Jakob, B., ‘Poncelet 5-gons and abelian surfaces’, Manuscripta Math. 83 (1994) no. 2, 183–198.CrossRefGoogle Scholar
Kumar, A. and Mukamel, R. E., 2014 Algebraic models and arithmetic geometry of Teichmüller curves in genus two’, Preprint, 2014, arXiv:1406.7057.Google Scholar
McMullen, C. T., ‘Billiards and Teichmüller curves on Hilbert modular surfaces’, J. Amer. Math. Soc. 16 (2003) 857–885.Google Scholar
McMullen, C. T., ‘Dynamics of SL2(ℝ) over moduli space in genus two’, Ann. of Math. (2) 165 (2007) 397–456.Google Scholar
Sakai, Y., ‘Poncelet’s theorem and curves of genus two with real multiplication of 𝛥 = 5’, J. Ramanujan Math. Soc. 24 (2009) no. 2, 143–170.Google Scholar
Sakai, Y., ‘Construction of genus two curves with real multiplication by Poncelets theorem’, Dissertation, Waseda University, 2010.Google Scholar
Smith, B. A., ‘Explicit endomorphisms and correspondences’, PhD Thesis, University of Sydney, 2005.Google Scholar
van der Geer, G., Hilbert modular surfaces , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 16 (Springer, Berlin, 1988).Google Scholar
van Wamelen, P. B., ‘Proving that a genus 2 curve has complex multiplication’, Math. Comp. 68 (1999) no. 228, 1663–1677.Google Scholar
van Wamelen, P. B., ‘Computing with the analytic Jacobian of a genus 2 curve’, Discovering mathematics with Magma (Springer, Berlin, 2006) 117–135.Google Scholar
van Wamelen, P. B., ‘Poonen’s question concerning isogenies between Smart’s genus 2 curves’, Math. Comp. 69 (2000) no. 232, 1685–1697.CrossRefGoogle Scholar
Wilson, J., ‘Curves of genus 2 with real multiplication by a square root of 5’, DPhil Thesis, Oxford University, 1998.Google Scholar
Wilson, J., ‘Explicit moduli for curves of genus 2 with real multiplication by Q(√5)’, Acta Arith. 93 (2000) no. 2, 121–138.Google Scholar
Wright, A., ‘Translation surfaces and their orbit closures: An introduction for a broad audience’, EMS Surv. Math. Sci. 2 (2015) 63–108.Google Scholar
Zorich, A., ‘Flat surfaces’, On random matrices, zeta functions and dynamical systems , Frontiers in Number Theory, Physics and Geometry 1 (Springer, Berlin, 2006) 439–586.Google Scholar