LECTURE 5. CONVERSE THEOREMS 159it would take us too far afield to present this in this setting, Lafforgue needed this
formulation in his proof of the Global Langlands Correspondence for GLn in this
context [59]. In Appendix B of that paper Lafforgue reformulated our Theorem
5.1 in terms of L-functions as rational functions and showed how to modify our
proof above to apply in this context. We recommend Lafforgue's Appendix B to
the interested reader.5.8. Conjectures
What are the optimal statements that one could hope for in a Converse Theorem?
At this point in time there seem to be two more or less accepted conjectures [12].
The first is credited to Jacquet. It assumes the least amount of twisting one
could hope for and still be able to control the cuspidality of II. The heuristics for
this conjecture can be found in the last section of [12]. Notations are as in Section
5.1.
Conjecture 5.1. Let II be an irreducible admissible representation of GLn(A) as
above. Let S be a finite set of finite places of k. Suppose that L(s, II x n') is nice
for all n' E Ts ( [ i]).
(1) If S = 0 then II is a cuspidal automorphic representation.
(2) If S -1- 0 then II is quasi-automorphic in the sense that there is an auto-
morphic representation II' such that IIv '.::'.II~ for all v ~ S.
The most ambitious conjecture we know of is due to Piatetski- Shapiro and is
explained in [12].
Conjecture 5.2. Let II be an irreducible admissible representation of GLn(A) as
above. Suppose that L(s,II©w) is nice for all w E T(l), that is, for all idele class
characters w. Then II is quasi-automorphic in the sense that there is an automorphic
representation II' such that IIv '.::'. II~ for all finite places of k where both II and II'
are unramified and such that L( s, II©w) = L( s, II' ©w) and c( s, II©w) = c( s, II' ©w)
for all w.
This conjecture would have many applications to number theoretic questions.
We refer the reader to Taylor's recent ICM talk for a discussion of some of these [93].