Evaluating modular forms on Shimura curves
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- by Paul D. Nelson;
- Math. Comp. 84 (2015), 2471-2503
- DOI: https://doi.org/10.1090/S0025-5718-2015-02943-3
- Published electronically: January 23, 2015
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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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Bibliographic Information
- Paul D. Nelson
- Affiliation: EPFL, Station 8, CH-1015 Lausanne, Switzerland
- Address at time of publication: ETH Zurich, Raemistrasse 101, 8092 Zurich, Switzerland
- Email: Paul.nelson@math.ethz.ch
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2471-2503
- MSC (2010): Primary 11F11, 11Y40; Secondary 11F27
- DOI: https://doi.org/10.1090/S0025-5718-2015-02943-3
- MathSciNet review: 3356036