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On the computation of coefficients of modular forms: The reduction modulo $ p$ approach


Authors: Jinxiang Zeng and Linsheng Yin
Journal: Math. Comp. 84 (2015), 1469-1488
MSC (2010): Primary 11F30, 11G20, 11Y16, 14Q05, 14H05
Published electronically: October 28, 2014
MathSciNet review: 3315517
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Abstract: In this paper, we present a probabilistic algorithm to compute the coefficients of modular forms of level one. Focusing on Ramanujan's tau function, we give the explicit complexity of the algorithm. From a practical viewpoint, the algorithm is particularly well suited for implementations.


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Additional Information

Jinxiang Zeng
Affiliation: Department of Mathematical Science, Tsinghua University, Beijing 100084, People’s Republic of China
Email: cengjx09@mails.tsinghua.edu.cn

Linsheng Yin
Affiliation: Department of Mathematical Science, Tsinghua University, Beijing 100084, People’s Republic of China
Email: lsyin@math.tsinghua.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-2014-02892-5
Keywords: Modular forms, Hecke algebra, modular curves, elliptic curves, Jacobian
Received by editor(s): May 17, 2013
Received by editor(s) in revised form: September 17, 2013
Published electronically: October 28, 2014
Additional Notes: This work was partially supported by NSFC grant No.11271212.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.