# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Bump, D., Hoffstein, J.: Some conjectured relationships between theta functions and Eisenstein series on the metaplectic group. In: Chudnovsky, D.V. et al. (eds.) Proceedings of the New York Number Theory Seminar. (Lect. Notes Math., vol. 1383, pp. 1-11) Berlin Heidelberg New York: Springer 1989 · Zbl 0677.10021 [2] Deligne, P.: Sommes de Gauss cubiques et rev?tements de SL(2). (Lect. Notes Math., vol. 770, pp. 244-277), Berlin Heidelberg New York: Springer 1980 [3] Heath-Brown, D.R., Patterson, S.J.: The distribution of Kummer sums at prime arguments. J. Reine Angew. Math.310, 111-130 (1979) · Zbl 0412.10028 [4] Hoffstein, J., Rosen, M.: Average values ofL-series in function fields. J. Reine Angew. Math. (to appear) · Zbl 0754.11036 [5] Kazhdan, D.A., Patterson, S.J.: Metaplectic forms. Publ. Math., Inst. Hautes ?tud. Sci.,59, 35-142 (1984) · Zbl 0559.10026 [6] Kubota, T.: On automorphic forms and the reciprocity law in a number field. Tokyo: Kinokuniya Book Store Co. 1969 · Zbl 0231.10017 [7] Patterson, S.J.: A cubic analogue of the theta series. J. Reine Angew. Math.296, 125-161 (1977) · Zbl 0358.10011 [8] Patterson, S.J.: Wittaker models of generalized theta series. In: Bertini, M.-J., Goldstein, C. (eds.) S?m. de th. des nombres. Paris, 1982-83. Boston: Birkh?user 1984 [9] Shimura, G.: On modular forms of half-integral weight. Ann. Math.97 (1973) · Zbl 0266.10022 [10] Suzuki, T.: Some results on the coefficients of the biquadratic theta series. J. Reine Angew. Math.340, 70-117 (1982) · Zbl 0499.10026 [11] Suzuki, T.: Rankin-Selberg convolutions of generalized theta series. J. Reine Angew. Math. (to appear) · Zbl 0733.11017 [12] Weil, A.: Sur certaines groupes d’operateurs unitaire. Acta Math.111, 143-211 (1964) · Zbl 0203.03305 [13] Zagier, D.: The Rankin-Selberg method for automorphic functions which are not of rapid decay. J. Fac. Sci. Univ. of Tokyo, Sect. I A,28, 415-437 (1981) · Zbl 0505.10011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.