International Congress of Mathematicians

122 J. W. Cogdell I. I. Piatetski-Shapiro3. Functoriality via the Converse TheoremIn order to apply these theorems, one must be able to control the analyticproperties of the F-function. However the only way we have of controlling global F-functions is to associate them to automorphic forms or representations. A minute'sthought will then convince one that the primary application of these results will beto the lifting of automorphic representations from some group H to GL n .Suppose that H is a reductive group over k. For simplicity of exposition we willassume throughout that H is split and deal only with the connected component of itsF-group, which we will (by abuse of notation) denote by L H [1]. Let n = ®'n v be acuspidal automorphic representation of H and p a complex representation of L H. Tothis situation Langlands has associated an F-function L(s,n,p) [1]. Let us assumethat p maps L H to GL n (C). Then by Langlands' general Principle of Functorialityto n should be associated an automorphic representation II of GL n (A) satisfyingL(s,U) = L(s,n,p), e(s,U) = e(s,n,p), with similar equalities locally and for thetwisted versions [1]. Using the Converse Theorem to establish such liftings involvesthree steps: construction of a candidate lift, verification that the twisted F-functionsare "nice", and application of the appropriate Converse Theorem.1. Construction of a candidate lift: We construct a candidate lift II = C^'n^on GL n (A) place by place. We can see what n^ should be at almost all places.Since we have the arithmetic Langlands (or Hecke-Frobenius) parameterization ofrepresentations n v of H(k v ) for all archimedean places and those non-archimedeanplaces where the representations are unramified [1], we can use these to associateto n v and the map p v : L H V —¥ L H —t GL n (C) a representation n^ of GL n (fc„). Thiscorrespondence preserves local F- and e-factorsL(s,U v ) = L(s,n v ,p v ) and e(s,U v ,ip v ) = e(s,n v ,p v ,ip v )along with the twisted versions. If H happens to be GL TO or a related group thenwe in principle know how to associate the representation n^ at all places now thatthe local Langlands conjecture has been solved for GL TO . For other situations, wemay not know what n^ should be at the ramified places. We will return to thisdifficulty momentarily and show how one can work around this with the use of ahighly ramified twist. But for now, let us assume we can finesse this local problemand arrive at a global representation II = C^'n^ such thatL(s,U) = JjL(s,n„) = Y[L(s,n v ,p v ) = L(s,n,p)and similarly e(s,U) = e(s,n,p) with similar equalities for the twisted versions. IIshould then be the Langlands lifting of n to GL n (A) associated to p.2. Analytic properties of global L-functions: For simplicity of exposition, let usnow assume that p is simply a standard embedding of L H into GL„(C), such as willbe the case if we consider H to be a split classical group, so that L(s, n, p) = L(s, n)is the standard F-function of n. We have our candidate II for the lift of n to GL„from above. To be able to assert that the II which we constructed place by placeis automorphic, we will apply a Converse Theorem. To do so we must control thetwisted F-functions L(s, II x n') = L(s,n x n') for n' £ T with an appropriate