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Diagonalizing Hilbert
cusp forms
Timothy Atwill |
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Vol. 169 (1995), No. 1, 33–49
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Abstract |
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We develop an operator Cq(Ψ𝒬) on the space 𝒮k(𝒩,Ψ) of Hilbert cuspforms as an alternative to the Hecke operator Tq for primes q dividing 𝒩. For f ∈𝒮k(𝒩,Ψ) a newform, we have f∣Cq(Ψ𝒬) = f∣Tq. We are able to decompose the space 𝒮k(𝒩,Ψ) into a direct sum of common eigenspaces of {Tp,Cq(Ψ𝒬) : p ∤ 𝒩,q∣𝒩}, each of dimension one. Each common eigenspace is spanned by an element with the property that its eigenvalue with respect to Tp (resp. Cq(Ψ𝒬)) is its p-th (resp q-th) Fourier coefficient. We finish by deriving bounds for the eigenvalues of Cq(Ψ𝒬). |
Mathematical Subject Classification 2000
Primary: 11F25
Secondary: 11F11, 11F41
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Milestones
Received: 16 November 1992
Revised: 10 January 1993
Published: 1 May 1995
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Authors |
| Timothy Atwill | |
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