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Non-existence of singular cusp forms. (English) Zbl 0768.11017

Let \(V\) be a finite dimensional vector space over (i) a global field \(k\) of characteristic not equal to 2, (ii) a quadratic extension of \(k\), or (iii) a quaternion algebra with center \(k\). Endow \(V\) with a nondegenerate sesquilinear form and let \(G\) be the isometry group of the form. The main theorem proved in the paper says that for any smooth function \(f\) in the space of a cusp form in \(L^ 2(G(k)/G(A))\) there is a nonzero Fourier coefficient of \(f\) along a unipotent subgroup with a suitably chosen character. From this theorem one can conclude that singular automorphic forms can not be cuspidal.
A special case of the above theorem for the symplectic group \(G\) was proved in a previous paper of the author [Duke Math. J. 59, 175-189 (1989; Zbl 0689.10041)].
Reviewer: Y.Ye (Iowa City)

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F12 Automorphic forms, one variable

Citations:

Zbl 0689.10041
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References:

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