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Automorphic Forms on GL(2) AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS Contents 1. Introduction and motivation 2 2. Representation Theoretic Notions 3 2.1. Preliminaries on adeles, groups and representations 3 2.2. Some categories of GF -modules and HF -modules 3 2.3. Contragredient representation 8 3. Classification of local representations: F nonarchimedean 9 3.1. (π; V ) considered as a K-module 10 3.2. A new formulation of smoothness and admissibility 10 3.3. Irreducibility 11 3.4. Induced representations 13 3.5. Supercuspidal representations and the Jacquet module 14 3.6. Matrix coefficients 17 3.7. Corollaries to Theorem 3.6.1 20 3.8. Composition series of B(χ1; χ2) 22 3.9. Intertwining operators 26 3.10. Special representations 36 3.11. Unramified representations and spherical functions 39 3.12. Unitary representations 44 3.13. Whittaker and Kirillov models 45 3.14. L-factors attached to local representations 57 4. Classification of local representations: F archimedean 72 4.1. Two easier problems 72 4.2. From G-modules to (g;K)-modules 73 4.3. An approach to studying (gl2;O(2; R))-modules 77 4.4. Composition series of induced representations 78 4.5. Hecke algebra for GL2(R) 81 4.6. Existence and uniqueness of Whittaker models 82 4.7. Explicit Whittaker functions and the local functional equation 84 4.8. GL2(C) 88 5. Global theory: automorphic forms and representations 88 5.1. Restricted products 88 5.2. The space of automorphic forms 89 5.3. Spectral decomposition and multiplicity one 90 5.4. Archimedean parameters 92 5.5. Classical forms on H 93 5.6. Hecke theory and the converse theorem 94 Date: Spring 2008. 1 2 AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS References 98 These notes were derived primarily from a course taught by Yannan Qiu at the University of Wisconsin in Spring of 2008. Briefly, the course was an introduction to the book of Jacquet and Langlands on Automorphic forms on GL2. I am greatly in debt to Yannan for teaching the class, and to Rob Rhoades who typed most of the original lecture notes. Additional details beyond Rob's notes (including solutions to many of the exercises) are included, and some additions and restructuring of the material has been made. The main reference for the course (and these notes) is Jacquet and Langlands book [5]. Some other references are Bump's book [2] and Jacquet's book [3]. Bump's book is easier to read but the real material is in [5]. Jacquet's book develops theory of GLn automorphic forms. 1. Introduction and motivation This course will be about L-functions and automorphic forms. There are two sorts of L-functions. (1) Artin L-functions: Suppose E=F is a finite extension of number fields and Gal(E=F ) acts on a finite dimensional vector space over C by ρ. Then we get L(s; ρ). (2) Automorphic L-function: Let G be a reductive group over F and π an automorphic representation. Then we get L(s; π). The Langlands Philosophy says that every Artin L-function is automorphic. More precisely, if L(s; ρ) is the Artin L-function corresponding to a non-trivial irreducible finite dimensional Galois representation, then there exists π, a cuspidal automorphic representation of GLn, such that L(s; ρ) = L(s; π). Remark. The full Langlands conjectures says more than just what is above, but they predict that irreducible ρ correspond to cuspidal π. The Langlands philosopy is useful because there are many analytic techniques that can be used to study automorphic L-functions. In particular, they are known to have holomorphic continuation and other such properties. However, many con- jectures (including Artin's conjecture) of number theory would be true if the same properties could be shown to hold for Artin L-functions. Remark. Artin's conjecture is that Artin L-functions are analytic (i.e. holomorphic on the entire complex plane.) It is known that they admit a meromorphic continua- tion, but it appears that the best way to prove holomorphicity is via the Langlands philosophy. The truth of the Langlands philosophy has been shown in the \GL1 case." The 1-dimensional Artin L-functions arise from characters ρ : Gal(E=F ) ! C×. By class field theory, we have ab 0 × × × Gal(E=F ) ! Gal(E=F ) = Gal(E =F ) ' AF =F Nm(AE0 ); 0 for some field E . We construct the automorphic representation of GL1 over F to be × × × × χ : AF =F Nm(AE0 ) ! C : AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS 3 × (Note that GL1(AF ) = AF .) It is true that L(s; ρ) = L(s; χ). The study of L(s; χ) can be found in Tate's thesis[7]. As in the example above, given a Galois representation, the first goal to proving the truth of Langlands philosopy would be to associate to it an automorphic rep- resentation. Then, one would want to show that the corresponding L-functions are the same. The primary goal of this course will be to explain what automorphic L-functions and automorphic representations are. 2. Representation Theoretic Notions 2.1. Preliminaries on adeles, groups and representations. General Set Up: Let G be a group over a global field F (which will typically be Q.) Possible examples include GLn, Sp2m, O(n). In the case of Sp2m we mean that the skew symmetric form is defined over F . To such a group, the notion of an automorphic representation can be defined. We take G = GL2. Then the adelic points of G can be viewed as the restricted product Y∗ G(AF ) = G(Fv) v places of F with respect to the subgroups Kv = GL2(OFv ). That is, (gv) 2 G(AF ) if and only 1 if gv 2 Kv for almost all v . Kν is a maximal compact subgroup of G(Fν ). This is a generalization of the adeles × Y × AF = Fv : v Here, the restricted product is taken with respect to the valuation rings OFν ⊂ Fv. Example 2.1.1. Y∗ × = × × ×: AQ R Qp p prime × Note that GL1(Zp) = Zp : Exercise 2.1.2. Show that GL2(Zp) is a maximal compact subgroup of GL2(Qp). It is a fact that if π is an irreducible automorphic representation of GL2(AF ) then it is the restricted tensor product π = ⊗πv, where πv are irreducible representations Q of GL2(Fv). Attached to each πv is an L-factor L(s; πv), and L(s; π) = L(s; πv) converges absolutely when s 0, and it has mermomorphic continuation. We do not describe what the restricted tensor product means at this time, but instead begin by classifying all local representations πv. We often treat the archimedean and nonarchimedean cases separately. 2.2. Some categories of GF -modules and HF -modules. In this section, we consider representations of GF = GL2(F ), where F is a non-archimedean local field. 4 F is a topological field so GF inherits the subspace topology from M2(F ) = F , and it is totally disconnected. A representation (π; V ) of GF is a complex vector space V together with an action of GF denoted π(g). In other words, V is a GF -module. We may also refer to the representation as π (when V is clear from context), or we may just say V is a GF -module (when the action π is clear from context.) 1i.e for all but finitely many. 4 AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS 2.2.1. Finite dimensional representations. Usually V will be infinite dimensional, but, in this section, suppose V is finite dimensional. Proposition 2.2.1. If (π; V ) is a continuous representation of GF , then ker(π) must contain an open subgroup. For example, GL2(OF ) is an open subgroup. × × Proof. We first prove the GL1 case. Say π = χ : Qp ! GL1(C) = C , this is the case that V is one dimensional. So the preimage of 1 equals ker(χ). Let R be a small open set around 1 in C, then χ−1(R) ⊇ N, N an open normal subgroup and χ(N) ⊂ R. Since N is a group, χ(N) must be a subgroup of C×. For R sufficiently small, the only such group is f1g. Hence, χ(N) = f1g, and N ⊆ ker(χ). This proof readily generalizes to GLn for any n. It is a fact from complex Lie theory, that any open neighborhood of the identity generates the connected component of the identity in the group. The topologies of C and Qp are not compatible, and that is what makes this × possible. C has an archimedean topology and Qp is totally disconnected. Remark. An open problem is to understand finite dimensional p-adic vector space representations which are much more complicated. 2.2.2. Smoothness and admissibility. Proposition 2.2.1 is not true in general when V is infinite dimensional. In the spirit of the proposition, and to obtain more manageable representations, we will restrict ourselves to the class of representations satisfying one or more of the following conditions. (A) (π; V ) is such that for any vector v 2 V , fg 2 GF j π(g)v = vg contains an open subgroup. (B) For all N ⊂ GF open, the set fv 2 V : Nv = vg is finite dimensional If (π; V ) satisfies (A), we call π smooth, and if it satisfies by (A) and (B), we say it is admissible. There is a good theory of admissible representations because it often reduces to that of finite groups where one can use orthogonality conditions and Schur's lemma.
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