Fourier coefficients of forms of CM-type

N. Laptyeva () and V. Kumar Murty ()
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N. Laptyeva: University of Toronto
V. Kumar Murty: University of Toronto

Indian Journal of Pure and Applied Mathematics, 2014, vol. 45, issue 5, 747-758

Abstract: 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.

Keywords: Cusp form; complex multiplication; Fourier coefficients (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s13226-014-0086-3

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