8 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN

integral that we do not discuss in detail. This is the fundamental relation that

Goldfeld-Hoffstein exploited in [36] to obtain asymptotics for

] P L(2s,Xm).

0±mX

Later Goldfeld-Hoffstein-Patterson [37] used similar Eisenstein series over an imag-

inary quadratic field together with the Asai integral [2] to get similar results for

L-functions attached to CM elliptic curves, and then Hoffstein and Rosen [40] used

the method over the rational function field ¥q(T).

Goldleld and Hoffstein anticipated the difficulty of generalizing this construc-

tion to autornorphic L-functions of higher degree. They write [36]:

At present, however, we cannot obtain mean value theorems for

quadratic twists of an arbitrary L-function associated to an auto-

rnorphic form... These appear to be difficult problems and their

solution may ultimately involve the analytic number theory of

GL(n).

2.3. Examples of multiple Dirichlet series arising from Rankin-Sel-

berg integrals. The Mellin transform and Asai integral mentioned above are

examples of Rankin-Seiberg integrals. In fact there are many other examples of

Rankin-Selberg integrals that give rise to multiple Dirichlet series. A number of

interesting examples can be understood as follows: in Section 2.2 we saw that the

Mellin transform, which gives a standard L-function if applied to a GL(2) form of

integral weight, gives a multiple Dirichlet series of the desired type when applied

to an Eisenstein series of half-integral weight. Note that the integral is no longer

an Euler product in that case. In a similar way we can look at other integrals that

give Euler products—Rankin-Selberg integrals—when applied to an autornorphic

form. Replacing the autornorphic form by a metaplectic Eisenstein series (like the

half-integral weight Eisenstein series 22), one can hope that the resulting object is

an interesting multiple Dirichlet series. We mention a few cases in which this hope

is realized.

2.3.1. Examples:

(1) Let ix be an autornorphic representation of GL(2) over Q(i). In [14] Bump,

Friedberg, and Hoffstein use ix to construct a metaplectic Eisenstein series En on

the double cover of GSp4. Now, an integral transformation due to Novodvorsky [50]

produces the spin L-function when applied to a non-metaplectic autornorphic form

on GSp4. When the same transformation is applied to the metaplectic Eisenstein

series En a multiple Dirichlet series of type (2.1) is created, with n = r = 2. The

choice of ground field was for convenience. A cleaner approach was found using

Jacobi modular forms and presented in [15], over ground field Q. For applications

to elliptic curves see [13]. Another construction of Friedberg-Hoffstein [33] obtains

this same multiple Dirichlet series using a Rankin-Selberg convolution of TX with a

half-integral weight Eisenstein series on GL(2). That paper works over an arbitrary

number field.

(2) Let ix be a GL(3) autornorphic form. Work of Bump, Friedberg, Hoffstein,

and Ginzburg (unpublished) obtains the double Dirichlet series of (2.1) as an in-

tegral of an Eisenstein series on the double cover of GSp6, or as an integral of an

Eisenstein series on SO (7) (these two groups are linked by the theta correspon-

dence) .