On the Fourier Coefficients of Biquadratic Theta Series

The concepts of a generalized theta function and of a metaplectic cover of GL2 were introduced by T. Kubota in the 1960s. This theory is concerned with a remarkable class of automorphic forms closely related to the nth-order reciprocity law over an algebraic number field k containing the nth roots of unity. A foundational discussion of the theory for the case of n-fold covers of GLr (r 2* 2) is given in [4]. The finer details of the theory of generalized theta functions are by no means understood, even over JGL2. In the case of a 3-fold cover of Q(V—3) one has a complete description [9]; in the case of more general 3-fold covers there is one undetermined parameter [4, p. 135]. In the case of a 2-fold cover of GL2 one recovers the classical theta series. These results can be generalized to an n-fold cover of GLr where they help in studying the Fourier coefficients of a theta function on an n-fold cover of GL2 for nSM. Several years ago the second author formulated a conjecture about the Fourier coefficients of the theta functions on a 4-fold cover of GL^; henceforth we shall call these 'biquadratic theta series'. This conjecture was supported by numerical evidence provided by N. Stephens but as no really efficient algorithms existed at that time the evidence was rather imprecise. The details were circulated privately and in [11] an attempt was made to formulate this conjecture in a general setting. This attempt was unsuccessful and the formulation given there is inconsistent. A discussion of the original motivation for the conjecture as well as a wide-reaching generalization can be found in [1]. Our objectives here are two-fold. First of all we shall give a precise version of the conjecture over any algebraic number field containing the fourth roots of unity. We are indebted to D. A. Kazhdan for his insistence on the necessity of such a formulation. This is given as Conjecture 2.11. One should note in this connection that the p{r, TJ) defined immediately before Theorem 1.9 are the Fourier coefficients of a generalized theta function but that we do not use this interpretation here. One consequence of our analysis is that one sees that the conjecture is special to this case and it gives no insight into the possible arithmetic nature of the Fourier coefficients of theta series for covers of degrees greater than 4. The second objective is to give numerical evidence for the correctness of the conjecture. The quantities in question are defined as the residues of certain Dirichlet series and as such are not easy to compute. In the original version of the conjecture, evidence had been computed by Nelson Stephens but as this was based on a Tauberian theorem the accuracy was low and could not be estimated