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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Evaluating modular forms on Shimura curves
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by Paul D. Nelson PDF
Math. Comp. 84 (2015), 2471-2503 Request permission

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.

References
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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
  • 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