where we fix TT and vary the twist by a character x; \ w^ range over the idele
class characters of order exactly n. We may also wish to modify the problem, and
suppose instead that x ranges over the subset of idele class characters of order
exactly n with local factors Xv specified at a finite number of places.
There are several natural questions to ask about this set of L-functions. The
first is nonvanishing:
(1) Given a point in the critical strip SQ (with 0 5ft(so) 1), can one show
there exist infinitely many \
a s
above with L(so,
n x
x) 7^ 0? This ques-
tion goes back to Shimura [54], Rohrlich [51], and Waldspurger [58]. A
particularly interesting choice is so \ For example, L-series associated
to elliptic curves of positive rank will conjecturally vanish at so = \ but
twists may not.
(2) If n = 2 (the case of quadratic twists) and TT is self-contragredient, and
if e(|,7r x x) ~ ~1 for all twists \ under consideration, can one show
there exist infinitely many x
s u c n
that £'(|? TT X X) ¥" 0? Note that under
these hypotheses, the functional equation guarantees a zero of odd order
for each twisted L-function at the center of the critical strip.
In these questions, we need not assume that TT is cuspidal - indeed, L(s, TT) could
be a product of L-series for lower-rank groups. Then the first question becomes
that of establishing a simultaneous non-vanishing theorem. Even in the case of two
independent GL(2) holomorphic modular forms, it is not known that there exists a
single quadratic twist such that both twisted L-functions are nonzero at the center
of the critical strip. In the case of two modular forms of weight 2, such a statement
would imply that given two elliptic curves E\, Ei over Q there exists a fundamental
discriminant D such that both twists and have finite Modell-Weil groups;
this is not presently known. Using multiple Dirichlet series, in fact one can establish
simultaneous non-vanishing for points so in the critical strip but sufficiently far from
the center of the strip [22] (see Theorem 6.1 in Section 6.2 below). Such results
can also be proved by the large sieve inequality, but the advantage of the multiple
Dirichlet series method is that the interval of nonvanishing obtained is independent
of the base field.
A related question, in some sense sharper, is to ask about the distribution of
twisted L-values. That is, one can seek to study the distribution of L(s, TT X X)
as we vary x a s above. For example, for positive integers k and weighting factors
a(5,7r, d) we can study the asymptotics of the moments
Y^ L(s,7rxx)fca(s,7r,d).
Given TT and &;, Langlands' theory of Eisenstein series implies that there is an
isobaric automorphic representation 11^ such that L(s,Uk x x) L(S,TT X x)k- So
it is natural to focus on the first moment, but to take TT to be general. Establishing a
suitable mean-value theorem for such moments would imply the Lindelof hypothesis
in the d-aspect.
Given a collection of interesting numbers a(d), the idea of studying their as-
sociated Dirichlet series Yl
is well-known. In the questions above, the
interesting numbers are themselves Dirichlet series: a(d) L(s, TT X \d)- Here Xd
(or Xd w n e n w e need to indicate the cover) is the character given by the n-th
power residue symbol Xd(a) = (§) , and is attached by class field theory to the
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