Euler products associated to metaplectic automorphic forms on the 3-fold cover of (4)

If ¢ is a generic cubic rnetaplectic form on GSp(4), that is also an eigenfunction for all the Hecke operators, then corresponding to X is an Euler product of degree 4 that has a functional equation and rnerornorphic continu.ation to the whole cornplex plane. This correspondence is obtained by convolving (b with the cubic 0-function on GL(3) in a Shirnura type RankinSelberg integral. 0. INTRODUCT[ON Suppose 0 is a metaplectic automorphic form of minimal level on the 3-fold cover of GSp(4) that is an eigenfunction of all the Hecke operators. If 0 has any non-zero Whittaker coefficients, then 0 is called generic. In this case, this paper will show tha1; there is a Dirichlet series in the Whittaker coefficients of 0 that has a formula1;ion as a degree 4 Euler product. Moreover, this Euler product has a meromorphic continuastion to the whole complex plane. This association of the Euler product with 0 will be obtained via a Shimura type JRankin-Selberg integral involving 0 and a H-function on the 3-fold cover of GL(3). Historically) the problem of associating an Euler product which has meromorphic continuation and functional equation with a metaplectic automorphic form originated with the work of Shimura [Shi]. More specifically, suppose f (z). = E a(n) qn is a holomorphic modular form of half-integral weight k/2, which is an eigenform of the Hecke operators Tp2, i.e. Tp2 f = )pf. Then via a JRankin-Selberg integral of the form f __ (0.1) | f (z) H(z) E(z, s) dz, where H(z) is a classical theta function and E(z, s) is an integral weight Eisenstein series, Shimura obtains an Euler product of the form (0.2) t| (1 _ Ap ps + pk-2-2s)-1 p The analytic continuation and functional equation of this Euler product follow from the siirlilar properties of E(z, s) in (0. 1) . Bump and Hoffstein [BH2] have subsequently extended these techniques of [Shig to GL(3) by finding a Rankin-Selberg integral of a metaplect1c automorphic form on the 3-fold cover of GL(3) which produces an Euler product of degree 3. Just as in [ShiX, this Euler product is shown to have meromorphic continuation to the whole Received by the editors November 14, 1995 and, in revised forrn, May 21, 1996. 1991 Mathematics S? lbject Classification. Prirnary llF55, llF30. @1998 Ameri( an Mathematical Society 975 This content downloaded from 157.55.39.45 on Wed, 05 Oct 2016 05:22:35 UTC All use subject to http://about.jstor.org/terms THOMAS GOETZE 976 complex plane and to have a functional equation under s 1-s. The integral which represents this Euler product involves the 0-function on the 3-fold cover of GL(3) over the field Q (e21rt/3) ? wllich has been studied independently by Proskurin [Pr] and Bump and HofEstein [BH12. In addition, Bump and HofEstein [BH2] have conjectured that Euler products with meromorphic continuation and functional equation may be obtained by convolving metaplectic automorphic forms on the n-fold cover of GL(r) against H-functions on the n-fold cover of GL(n). This was carried out in [BH3] in the case r-2 and n > 2. Friedberg and Wong [FrW] have also used Shimura's method to associate an Euler product to a generic metaplectic automorphic form on the double cover of the symplectic group GSp(4). They have found an integral (inspired by Novo dvorsky's GSp(4)xGL(2) convolution) involving ametaplectic automorphic form on the double cover of GSp(4), the H-function on the double cover of GL(2), and a (non-metaplectic) Eisenstein series on GL(2), that yields a degree 5 Euler product. This Euler product is shown to have meromorphic continuation and a functional equation, and furthermore it has the same local Euler factors as the L-function of an automorphic form on GSp(4). As in [Shi, BH2], the Euler product found by Friedberg and Wong is explicitly constructed from the Whittaker coeEcients of the metaplectic automorphic form. Alternatively, Flicker [Fli], Kazhdan and Patterson [KaP2], and Flicker and Kazhdan [FliKa] have used the trace formula to generalize [Shi] by showing that (in many situations) there exists a correspondence between metaplectic automorphic forms and (non-metaplectic) automorphic forms. Indeed: Shimura actually proves in [Shi] that the Euler product (0.2) is the L-function of a holomorphic integral weight modular form. In using the trace formula, however: explicit information about the interplay between the metaplectic Fourier coefficients and the corresponding L-functions (see (0.2)) is not obtained. If a generalized Shimura correspondence does exist between generic metaplectic and non-metaplectic automorphic forms, then the associated Euler products obtained by Bump-Hofistein, Friedberg-Wong, and this paper will be the L-functions of the corresponding non-metaplectic forms. There is evidence that the degree 4 Euler product obtained in this paper is the L-function of an automorphic form on GSp(4) Savin [Sa] has shown that there is an algebra isomorphism between the local Iwahori Hecke algebra of GSp(4) and the local Iwahori Hecke algebra on the 3-fold cover of GSp(4) This suggests that; if a Shimura correspondent exists in this situation, it should be an automorphic form on GSp(4)e Since there is a representation of degree 4 on the L-group of GSp(4): automorphic forms on GSp(4) will have natural L-functions with Euler products of degree 4 In this sense, having a degree 4 Euler product is consistent with Savin's results. The main results of this paper will be found in Theorems 3.1 and 5.1, which are summarized as follows: Mairl Theorem. Suppose Q is a generic metaplectic C? lsp form of minimal level on the S-fold cover of G8p(4) that is an eigenf? lnction of all the lTecke operators. Then there is a degree J E? ller prod? lct, with a meromorphic contin? ltation, which cctn be explicitly constr? lcted from the Whittaker coefficients of f0Je This association is realized as a Shim? lra type Rankin-Selberg integral of 0 against an Eisenstein series ind? lced from a f)-f? lnction on the S-fold cover of GL(3). This content downloaded from 157.55. 39.45 on Wed, 05 Oct 2016 05:22:35 UTC All use subject to http://about.jstor. org/terms