Appendix: Modular forms, the Ramanujan conjecture and the Jacquet-Langlands correspondence Jonathan D. Rogawski1) The theory developed in Chapter 7 relies on a fundamental result (Theorem 7 .1.1) asserting that the space L2(f\50(3) x PGLz(Op)) decomposes as a direct sum of tempered, irreducible representations (see definition below). Here 50(3) is the compact Lie group of 3 x 3 orthogonal matrices of determinant one, and r is a discrete group defined by a definite quaternion algebra D over 0 which is split at p. The embedding of r in 50(3) X PGLz(Op) is defined by identifying 50(3) and PGLz(Op) with the groups of real and p-adic points of the projective group D*/0*. Although this temperedness result can be viewed as a combinatorial state­ ment about the action of the Heckeoperators on the Bruhat-Tits tree associated to PGLz(Op). it is not possible at present to prove it directly. Instead, it is deduced as a corollary of two other results. The first is the Ramanujan-Petersson conjec­ ture for holomorphic modular forms, proved by P. Deligne [D]. The second is the Jacquet-Langlands correspondence for cuspidal representations of GL(2) and multiplicative groups of quaternion algebras [JL]. The proofs of these two results involve essentially disjoint sets of techniques. Deligne's theorem is proved using the Riemann hypothesis for varieties over finite fields (also proved by Deligne) and thus relies on characteristic p algebraic geometry. By contrast, the Jacquet­ Langlands Theorem is analytic in nature. The main tool in its proof is the Seiberg trace formula. Furthermore, Deligne's Theorem is conveniently expressed in the classicallanguage of modular forms, while the Jacquet-Langlands correspondence requires the language of infinite-dimensional representation theory. The purpose of this appendix is to explain the Statements of the Deligne's Theorem and the Jacquet-Langlands correspondence, briefly outline their proofs, and explain how they combine to yield Theorem 7.1.1. The standard references [AA], [Co], [D3], [G] should be consulted for further details and related topics. I would like to thank Victor Tan for carefully reading the manuscript and suggesting several corrections. 1) Partially supported by a grant from the NSF. 136 APPENDIX A.O Preliminaries In this section, we fix notation and review some basic facts. See [CF] for a general treatment of adeles and ideles and [GPPS], [G] for a discussion of adele groups. Throughout this appendix, p will denote a prime of 0 (the infinite prime p = oo is allowed). If p is finite, Op is the field of p-adic numbers and ~P is the ring of p-adic integers. For p = oo, 0 00 = R The absolute value on Op will be denoted by I lp· Let G denote the group GL(2) of invertible 2 x 2 matrices, and define the subgroups: The decomposition B = MN holds. The center of G is the group of scalar matrices Z={(~ ~)}· We shall use algebraic group notation. Thus, if H is one of the above groups and R is any ring, then H(R) is the group of R-valued points of H, i.e., the group of elements of H with entries from R. We will write Hp for H(Op)· For all p, Gp is a locally compact group under the topology inherited from the natural coordinate topology on the set of 2 x 2 matrices. For p = oo, G00 is a Lie group. For p < oo, Gp is totally disconnected, in the sense that there exists a basis for the open neighborhoods of the identity consisting of subgroups. Such a basis is given by the set of congruence subgroups 1 + pn M2 (~p) for n 2: 1. Let Kp be the following standardmaximal compact subgroup of Gp. If p < oo, Kp = GL(~p). For p = oo, let r(O) = ( co~ (} sin (} ) - smO cos(} and set Koo = {r(O) : (} E [0, 21r)}. The lwasawa decomposition holds for all p. As explained in Chapter 6, the adele ring A of 0 is the restricted direct product of the fields Op: A ~ { (ap) E I} o, 'a, E z, fo,.Wnost aU primes p < oo} . A.o PRELIMINARIES 137 Let At be the ring of finite adeles, defined as a restricted direct product as above but without the infinite factor R When convenient, we also identify At with the subring of elements (ap) E A suchthat a00 = 0. For any adele a, we write at for the finite part of a, i.e., the tuple obtained by dropping a00 • We view 0 as a subring of A relative to the diagonal embedding. Thus x E 0 is identified with the adele (x, x, ... ) .Let :z = l~7L/N = rr 1Lp. N p<oo The ring A has the stru<:.._ture of a locally compact topological ring for the topology relative to which ~ x 7L with its product topology is an open subring (cf. 6.1). The image of 0 in A is a discrete subgroup since 0 n ( -1, 1) x Z = { 0}. Furthermore, there is a surjective homomorphism of Z onto 7LjN with kemel NZ, and for all integers N we have the following decomposition (referred to as the Strong Approximation Theorem): A = 0 + (~ X NZ). It follows that 0\A = N7L\(~ x NZ) for all N and also that 0\A is compact. The idele group 0 is the group of invertible elements in A: I~ { (ap) EI} o; 'ap Ez; fo' alm"'t all prime' p < oo}. The group Ot of finite ideles is the group of invertible elements in At. Note that Z* = Tir<oo 7LP. The idele group is a locally compact group for the topology relative to which ~* x Z* with its product topology is an open subgroup. The image of 0* in 0 is a discrete subgroup since 0* n ~+ x Z* = { 1}. Here ~+ is the multiplicative group of positive real numbers and 0* is identified with its image under the diagonal embedding. For X E 0, the absolute value is defined as the product of the local absolute values: lxl = rr lxlr· p The product is actually finite since ixlr = 1 for almost all p. Let 01 be the group of ideles x such that lxl = 1. The product formula asserts that 0* is contained in 01. It follows from the unique factorization into primes that 0 decomposes as a direct product 0 = 0* ( ~~ X Z*) . (1.1) 138 APPENDIX In fact, for ~y idele x, define the rational number a = sgn(xoo) flp<oo lxpl· Then ax E lffi'f_ x ll.*. The adele group G(A) is the group of invertible 2 x 2 matrices with entries from Al: G(A) ~ { g ~ (&,) E l} G(Op): gp E G(Zp) fm almoot all p}. This is a locally compact group for the topology relative to which G(lffi) x flp<oo G(ll.p) with its product topology is an open subgroup. A Hecke character w is a continuous character of 0 which is trivial on Q*. Let Wp and Wf denote the restrictions of w to Q~ and lt, respectively. By (1.1), we may view w as a continuous character of lffi'f_ x Z*. For all s E C, x ___, lxl 5 is a Hecke character which is trivial on Z*. On the other hand, if w' is a Dirichlet character, i.e., a character of (7l./N)*, then w' defines a Hecke character w which is trivial on lffi'f_ via the homomorphism «reduction modulo N»: Z* ___, (7l./N)*. Observe that with this definition, if (p,N) =I, then wp(p) = w'(p)-1. lndeed, 1 = w((p,p,p, .. .)) = wp(p)w((p,p, ... , l,p, ... ) = wp(p)w'(p), where a = (p,p, ... ,l,p, ... ) is the idele suchthat ap = 1 and ap = p for all primes C =f=. p. Every Hecke character w decomposes uniquely as a product 5 w0 ( x) lx 1 , where wo is defined by a Dirichlet character. Note that Z (Al) is iso­ morphic to 0. We may therefore regard any Hecke character as a character of Z(O)\Z(A). A.I REPRESENTATION THEORY AND MODULAR FORMS 139 A.l Representation theory and modular forms A unitary representation of a topological group His a pair (1r, V), where Visa Hilbert space and 1r is a homomorphism from H to the group U(H) suchthat the map Hxv-v (h, v)- 1r(h)v is continuous. Sometimes, we write V1r for the space V, and at other time, we drop V from the notation and notate only the homomorphism 1r. If H is a unimodular locally compact group and H' is a closed unimodular subgroup of H, then there exists an H-invariant measure on the quotient space H'\H and we may define the Hilbert space L 2 (H'\H). The group H acts unitarily on L2 (H'\H) by right translation p: (p(h)<!J)(x) = <jJ(xh). In the theory of automorphic forms, we are interested in the representation of G(A) on L2(G(O)\G(A)). However, the quotient Z(O)\Z(A) is noncompact, and it is more convenient to consider a slightly different Hilbert space. Fix a unitary character w : Z(O)\Z(A) - C* and Iet L2(w) be the Hilbert space of functions </J on G(O)\G(A) such that <jJ(zg) = w(z)</J(g) and r I<P(g)l2 dg < 00 jG(Q)Z(A)\G(A) where dg is a G(A)-invariant measure on G(O)Z(A)\G(A).