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