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Fourier coefficients of forms of CM-type

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Abstract

Let f be a cuspidal normalized eigenform of weight ≥ 2 for Г0(N),with Fourier expansion

$f(z) = \sum\limits_{n = 1}^\infty {a_f (n)e^{2\pi inz} } $

While the Galois representations associated to f can be used effectively to study the divisibility properties of the Fourier coefficients, it is very difficult to analyze the condition a f (p) = 0 (mod p). In this paper, we show that the problem is accessible in the case that f has complex multiplication. Under some mild conditions on f, we show that for p sufficiently large, a f (p) = 0 (mod p) in fact implies that a f (p) = 0.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada

    N. Laptyeva & V. Kumar Murty

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  1. N. Laptyeva
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  2. V. Kumar Murty
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Correspondence to N. Laptyeva.

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Laptyeva, N., Kumar Murty, V. Fourier coefficients of forms of CM-type. Indian J Pure Appl Math 45, 747–758 (2014). https://doi.org/10.1007/s13226-014-0086-3

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  • DOI: https://doi.org/10.1007/s13226-014-0086-3

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