A special case of one of the results in the paper of Luo and Ramakrishnan
([45]) is
Theorem [45] Let /, g be two Hecke newforms for a congruence subgroup of
SL2{1). Suppose there exists a nonzero constant c s.t.
L(lJ®Xd) =cL(±,g®Xd)
for all quadratic characters Xd- Then f eg.
This theorem has an application to a question of Kohnen: let g\, gi be two
newforms in the Kohnen subspace S+
with Fourier coefficients fri(^), ^(n ) re-
spectively. Suppose
for almost all fundamental discriminants with (—l)kD 0. Then g\ = ±#2, i.e. you
can't just switch some of the signs of the coefficients and get another eigenform.
The proof uses Waldspurger's formula relating the square of bj(\D\) to a suitable
multiple of a twisted central value. A similar theorem holds for central derivatives
in the case of negative root number ([46]). By the theorem of Gross-Zagier, this
allows one to determine an elliptic curve by heights of Heegner points.
Recently, the results of Luo and Ramakrishnan have been extended in two
directions using the the methods of multiple Dirichlet series. First, Ji Li [44]
extends [45] to 7Ti, 712 cuspidal automorphic representations of GL2(A#), f°r K a n
arbitrary number field. Secondly, Chinta and Diaconu [21] extend [45] to symmetric
squares of cusp forms on GL2(AQ).
Both of these theorems are proved by considering twisted averages of twists
of central L-values. The result of J. Li should also extend to cover the case of
determining n by twisted central derivatives. Over a number field, the averaging
method employed by [45] (originating in the work of Iwaniec [42] and Murty-Murty
[49]) runs into complications.
By contrast with J. Li's result, the result of [21] is valid only over Q. This is
because the authors need to use the bound
(3.30) £ | L ( i , 7 r ®
X d
) | «
a : 5 / 4 + £
for 7r an automorphic form on GL(3). Of course,
is expected because of the Lindelof conjecture but this is far out of reach. The
proof in [21] of the bound (3.30) is valid only over Q, because of an appeal to a
character sum estimate of Heath-Brown [38]. It would be of great interest to see
what types of bounds can be proved over an arbitrary number field.
4. Higher Twists
In this section we discuss higher twists. The situation here is different from the
quadratic twist case due to epsilon factors.
Let 7r be an automorphic representation of GL(r) over given base field, and
for the moment let L(s, 7r) denote its standard complete L-function. Then L(s, n)
satisfies a functional equation
L(s, 7r) = e(s, n) L(l 5, n).
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