On Kubota’s Dirichlet Series

Kubota [19] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with n-th order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the precise form required by the companion paper [2]. Closely related results are in Eckhardt and Patterson [10]. The Kubota Dirichlet series are the entry point to a fascinating universe. Their residues, for example, are mysterious if n > 3, though there is tantalizing evidence that these residues exhibit a rich structure that can only be partially glimpsed at this time. When n = 4 the residues are the Fourier coefficients of the biquadratic theta function that were studied by Suzuki [23]. Suzuki found that he could only determine some of the coefficients. This failure to determine all the coefficients was explained in terms of the failure of uniqueness of Whittaker models for the generalized theta series by Deligne [9] and by Kazhdan and Patterson [15]. On the other hand, Patterson [22] conjectured that the mysterious coefficients are essentially square roots of Gauss sums. Evidence for Patterson’s conjecture is discussed in Bump and Hoffstein [6] and in Eckhardt and Patterson [10], where the conjecture is refined in light of numerical data. Partial proofs were given by Suzuki in [24] and [25]. Another set of conjectures relevant to the mysterious coefficients of n-th order theta functions were given by Bump and Hoffstein, who considered theta functions on the n-fold covers of GLr for arbitrary r. They are expressed as identities between Rankin-Selberg convolutions of generalized theta series and Whittaker coefficients of Eisenstein series on the metaplectic group, but they boil down to properties of the residues of Kubota Dirichlet series, and their higher rank generalizations. See Bump and Hoffstein [6], Bump [4] and Hoffstein [12]. These conjectures are different from the Patterson conjecture, and there are other considerations which suggest that there may be further unproved relations beyond those described in the conjectures of Patterson and Bump and Hoffstein.