Theta functions on the \(n\)-fold metaplectic cover of \(SL(2)\) — the function field case.

*(English)*Zbl 0761.11024This 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.

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.

Reviewer: S.J.Patterson (GĂ¶ttingen)

##### MSC:

11F30 | Fourier coefficients of automorphic forms |

11F27 | Theta series; Weil representation; theta correspondences |

##### Keywords:

metaplectic theta functions; \(n\)-fold cover of SL(2); Fourier coefficients; Eisenstein series; residues; rational function field; Rankin-Selberg integral; biquadratic theta series##### References:

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