6 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN

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 E® and E® 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).

cond(x)X

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

a(d)d~s

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