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Shimura Curve Computations Via K3 Surfaces of Néron–Severi Rank at Least 19

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Algorithmic Number Theory (ANTS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5011))

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Abstract

In [E1] we introduced several computational challenges concerning Shimura curves, and some techniques to partly address them. The challenges are: obtain explicit equations for Shimura curves and natural maps between them; determine a Schwarzian equation on each curve (a.k.a. Picard-Fuchs equation, a linear second-order differential equation with a basis of solutions whose ratio inverts the quotient map from the upper half-plane to the curve); and locate CM (complex multiplication) points on the curves. We identified some curves, maps, and Schwarzian equations using the maps’ ramification behavior; located some CM points as images of fixed points of involutions; and conjecturally computed others by numerically solving the Schwarzian equations.

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA 02138,  

    Noam D. Elkies

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  1. Noam D. Elkies
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Alfred J. van der Poorten Andreas Stein

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Elkies, N.D. (2008). Shimura Curve Computations Via K3 Surfaces of Néron–Severi Rank at Least 19. In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_13

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  • DOI: https://doi.org/10.1007/978-3-540-79456-1_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-79455-4

  • Online ISBN: 978-3-540-79456-1

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