International Congress of Mathematicians

120 J. W. Cogdell I. I. Piatetski-Shapiroadmissible representation II of GL„ (A) to be automorphic and cuspidal in terms ofthe analytic properties of Rankin-Selberg convolution F-functions L(s, II x n') of IItwisted by cuspidal representations n' of GL TO (A) of smaller rank groups.To use Converse Theorems for applications, proving that certain objects areautomorphic, one must be able to show that certain F-functions are "nice". However,essentially the only way to show that an F-function is nice is to have it associatedto an automorphic form. Hence the most natural applications of ConverseTheorems are to functoriality, or the lifting of automorphic forms, to GL n . Moreexplicit number theoretic applications then come as consequences of these liftings.Recently there have been several applications of Converse Theorems to establishingfunctorialities. These have been possible thanks to the recent advances inthe Langlands-Shahidi method of analysing the analytic properties of general automorphicF-functions, due to Shahidi and his collaborators [21]. By combiningour Converse Theorems with their control of the analytic properties of F-functionsmany new examples of functorial liftings to GL„ have been established. These aredescribed in Section 4 below. As one number theoretic consequence of these liftingsKim and Shahidi have been able to establish the best general estimates overa number field towards the Ramamujan-Selberg conjectures for GL 2 , which in turnhave already had other applications.2. Converse Theorems for GL nLet k be a global field, A its adele ring, and ip a fixed non-trivial (continuous)additive character of A which is trivial on k. We will take n > 3 to be an integer.To state these Converse Theorems, we begin with an irreducible admissiblerepresentation II of GL n (A). It has a decomposition II = C^'n^, where n^ is anirreducible admissible representation of GL n (fc„). By the local theory of Jacquet,Piatetski-Shapiro, and Shalika [9, 11] to each n^ is associated a local F-functionL(s,U v ) and a local e-factor e(s,U v ,ip v ). Hence formally we can formL(s,U) = JJ_L(s,U v ) and e(s,U,ip) = JJ_e(s,U v ,'ip v ).We will always assume the following two things about II:(1) L(s,U) converges in some half plane Re(s) >> 0,(2) the central character un of II is automorphic, that is, invariant under k x .Under these assumptions, e(s, II, ip) = e(s, II) is independent of our choice of ip [4].As in Weil's case, our Converse Theorems will involve twists but now by cuspidalautomorphic representations of GL TO (A) for certain m. For convenience, letus set A(m) to be the set of automorphic representations of GL TO (A), Ao(m) theset of (irreducible) cuspidal automorphic representations of GL TO (A), and T(m) =UrfLi Ao(d). If S is a finite set of places, we will let T s (m) denote the subset ofrepresentations n £ T with local components n v unramified at all places v £ S andlet Ts(m) denote those n which are unramified for all v $ S.