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Theta functions on the \(n\)-fold metaplectic cover of \(SL(2)\) — the function field case. (English) Zbl 0761.11024
This paper is devoted to a study of the metaplectic theta functions on an \(n\)-fold cover of SL(2). Kubota’s definition of such functions is as residues of Eisenstein series. The Eisenstein series and their Fourier coefficients are analytic functions with a known, but rather complicated, functional equation. In the case of a function field (over a finite field) the notion of ‘analytic’ is replaced by ‘rational’ and the location of the poles and the functional equation limit the degrees of the numerators; the denominators are known. Thus the calculation of the Fourier coefficients of the Eisenstein series, and their residues, become a finite one. In fact the scale of these calculation is daunting unless one treats a rational function field. This is what the author does here.
The calculations require a lot of ingenuity to keep them in the range that can be treated by hand. The author uses his results, which cover a large number of cases to make a conjecture about the nature of the Fourier coefficients in the number-field case as well, but at present there is very little evidence for this.
Finally the author uses the same principle again with a Rankin-Selberg integral (which is determined by a finite number of its coefficients as a Dirichlet series). In this way he is able to verify a conjecture made by the reviewer in the case of biquadratic theta series.

MSC:
11F30 Fourier coefficients of automorphic forms
11F27 Theta series; Weil representation; theta correspondences
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References:
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