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Evaluating modular forms on Shimura curves

Author: Paul D. Nelson
Journal: Math. Comp. 84 (2015), 2471-2503
MSC (2010): Primary 11F11, 11Y40; Secondary 11F27
Published electronically: January 23, 2015
MathSciNet review: 3356036
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Abstract: Let $ f$ be a newform, as specified by its Hecke eigenvalues, on a Shimura curve $ X$. We describe a method for evaluating $ f$. The most interesting case is when $ X$ arises as a compact quotient of the hyperbolic plane, so that classical $ q$-expansions are not available. The method takes the form of an explicit, rapidly-convergent formula that is well-suited for numerical computation. We apply it to the problem of computing modular parametrizations of elliptic curves, and illustrate with some numerical examples.

An important ingredient is a new method for numerically computing
Petersson inner products, which may be of independent interest.

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Additional Information

Paul D. Nelson
Affiliation: EPFL, Station 8, CH-1015 Lausanne, Switzerland
Address at time of publication: ETH Zurich, Raemistrasse 101, 8092 Zurich, Switzerland

Received by editor(s): October 9, 2012
Received by editor(s) in revised form: October 10, 2013
Published electronically: January 23, 2015
Additional Notes: The author was supported by NSF grant OISE-1064866 and partially supported by grant SNF-137488 during the completion of this paper.
Article copyright: © Copyright 2015 American Mathematical Society