Evaluating modular forms on Shimura curves

Nelson, Paul

doi:10.1090/S0025-5718-2015-02943-3

Evaluating modular forms on Shimura curves

Author: Paul D. Nelson
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
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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