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).
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
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) .
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