Eisenstein Series and the Selberg Trace Formula. Ii

The integral of the kernel of the trace formula against an Eisenstein series is investigated. The analytic properties of this integral imply the divisibility of the convolution L-function attached to a form by the zeta function of the field. Introduction. This paper is a sequel and generalization of [12], but can be read independently of that paper; in particular, we will repeat the description of the problem given in the introduction of [12], now, however, in an adelic setting. Let F be a global field, A its ring of adeles, and p0 the representation of PGL(2, A) by right translation on the space of cusp forms Lq(PGL(2, F)\ PGL(2,A)). Given any 1, obtaining the same formula as given above but without the term I5. The extra term Is(s) in the strip 0<^(5)<1 appears as a residue coming from the poles of Ix(s, u) at u = s/2 and « = 1 — 5/2 when x is the trivial character. (For the same reason, the poles at u = (1 + 5)/2 give a contribution which cancels the term IA(s) when we cross the line 3$e{s) = 0. A similar phenomenon already occurred in [12].) This will be carried out in §3, where we also give the main application—the divisibility of I(s) by $F{s), and as a consequence of this the holomorphy of the symmetric square of the L-series attached to a cusp form. This latter fact was proved for classical holomorphic forms by Shimura [10] and independently by one of the authors [11]; Shimura's method was generalized by Gelbart and Jacquet [3] to an adelic setting, while the present paper is essentially the adelic generalization of [11]. Another application is—or should be—the trace formula, which as explained above arises by calculating the residue of I(s) at 5 = 1. This calculation is complicated by the fact that some of the terms in the formula for I(s) (namely the terms Iv I2 and / in the theorem above) have double poles at s = 1, and although the coefficients of (5 l)"2 naturally cancel in the sum, this means that we need two terms of the Laurent expansion rather than just the leading term in order to calculate the residue. We were able to calculate the residues of IE and Ix to I5 explicitly, obtaining six of the seven terms in the usual adelic Selberg trace License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use EISENSTEIN series and the SELBERG FORMULA. II 3 formula (cf. 3.2), but the formula for Ix(s, u) is so complicated that we could not reduce the expression for its residue to the corresponding term in the trace formula as it is usually formulated. After almost a five year delay during which we hoped to return to and settle this point, we decided to publish the paper with the deduction of the usual trace formula from our formula left incomplete. In any case, both of these two consequences of our theorem—the trace formula and the holomorphy of symmetric square L-functions for representations of GL(2) —were already known. The main interest in our result lies in the form of the identity, which can be thought of as a generalization of the trace formula in which the various terms are expressed locally (i.e. as products of local integrals). Furthermore, the method can in principle be generalized to GL(«). For GL(3) this has already been partially carried out by Parameswaran Kumar. 0. Notations and conventions. Tate integrals. F denotes a global field, Fv the completion of F at a typical place v, Rv the valuation ring of FD, and A and A x the adeles and ideles of F. We will generally use Greek letters for elements of F and Latin letters for adelic variables. The norm map from A x to R x is denoted by | |, the set of ideles of norm 1 by A f. We choose once and for all a splitting A x = A x X R x and denote by A the set of characters on A X/Fx which are trivial on R X; thus the most general (quasi-) character on A X/Fx has the form a <-> x(« )MS witn X G A, 5 e C. We choose once and for all a nontrivial additive character \p: A/F -> C. The Haar measures on A, Af and A x are normalized by /A/f dx = 1, J/\*/F>< dxal = 1 and dxa = dxax X dxt, where dxt = dt/t is the standard Haar measure on R x. We denote by G the algebraic group GL(2) and by Z, A, N and P (= AN) the subgroups of matrices of the form (g °), (g °), (0 x) and (g £), respectively; we write GF, Gv and GA for G(F), G(FV) and G(A) and similarly for the other groups. We denote by K = II,, Kv the standard maximal compact subgroup of GA. We will often identify JVA, ZA and AA with A, A x and (A x)2, respectively; in particular, this will be done to define the Haar measures on these groups and also to identify i//, elements of A, and pairs of elements of A with characters of N^/NF, of Z^/ZF, and of A^/AF, respectively. The Haar measure on K is normalized so that fKdk = 1 and the Haar measure on GA then chosen so that f f(g)dg=( f f f(kna)dndadk= f f f f{kan)\a2/ax\dndadk JGA JK JAA JNA JK JAA JNA (here av a2 denote the diagonal components of a e A). We denote by w the element (° 0) of G and by g >-» gl (g e G) the involution g >-> 'g'1. We denote by y(A) the space of Schwartz-Bruhat functions and by yo(A) the subspace spanned by products $ = Ily^,, whose components at real and complex places have the form ®v(xv) = e'"*" X polynomial in xv (v real), (x)= f ${u)^{xu)du ($e^(A)), •'a $(x,y)=[ f ®(u,v)j(xu+ yv)dudv ($ey(A2)), •'A •'A (jc, v) = f <&(x,v)4>{yv)dv ($e^(A2)), •'a the Fourier transform and Fourier transform with respect to the second variable. For ge6A and $ <= y(A2), gO denotes the function g$(x, y) = $[(x, v)g]; thus g3> = |det g|_1g'. Finally, we call any integral of the form f <£>{a)x(a)\a\Sdxa (*e^(A),xeA,jGC,*(j)>l) a Tate integral for L(5, x) (where L(s, x) itself is defined by making the appropriate standard choice for $) and denote by L(0, 5, x) the meromorphic continuation of this function, i.e. L(0>,5,X) = L($,l-5,x) = [ (a)X(a)\a\Sdxa+ f $(a)x(a)\a\1~sdxa f*(ol_*(o) lfy = 1 + 5-1 5 X ' 10 otherwise. Finally, we mention the identity f f (0) ($e^(A2)), JK •'A* which can be thought of as the analogue of the ordinary polar coordinates formula f° f" (x + iy)dxdy= f2' f <&[rei9]r2 — d0 in R2. The constant c is given by L(2,lF)\D\V2 ResJ_1L(5,lF)' where D is the discriminant and L(s,lF) the zeta-function (with factors at infinity added) of F. It is also equal to \ vol( ZAGF\ GA). 1. Review of Eisenstein series and related topics. Almost all of the material in this section is standard; we refer the reader to [2, 4]. 1.1. Eisenstein series. For Xi, Xi G A and SeC we denote by irXltXltl the representation of GA by right translation on the space H(xi,X2>s) of (classes of) functions / on GA satisfying 4(o l)s] = Xi(a)x2(b)\^\f(g) («,6GAx,xeA,geGA) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use EISENSTEIN SERIES AND THE SELBERG FORMULA. II 5 and f \f(k)\2dk Xi>s) is a n^er bundle over C and we have trivialized this bundle. Given any / e H(xi, X2X tne corresponding section of U5 //(Xi, X2> -0 is defined by Sf[(o Xb)k's]=xMx2(b)\^f(k) (a,be/\x,x^J\,k^K) and the corresponding Eisenstein series (for 3le(s) > 1) by Ef(g,s)= E 5/(yg,5) (geGA). yePF\GF For later purposes, we fix once and for all an orthonormal basis (of AT-finite functions) {/a}„e^(x x ) of the Hilbert space #(xi>X2); tne corresponding functions Sfa(g, 5) and £/a(g, 5) will sometimes be denoted simply Sa(g, s), Ea(g, s). The definition of Eisenstein series just given corresponds to the classical series E(z,s)= E My*)* (z,iGC,Im(z)>0,*(i)>l). re{±(o ?)}\SL(2,Z) For analytic purposes another definition, analogous to the function ±Im(z)s Z (mz + n)-2s = U2s)E(z,s), (m,n)eZ2-0 is more convenient: For $ e S(A2) the function /(g.*,Xi.X2.*) = X1(detg)|detg|7 $[{0,t)g]XlXll(t)\t\2'd*t (which is a Tate integral for L(25,X1X21)) belongs to #(Xi>X2'-y) and tne corresponding Eisenstein series E(g,s) = E(g,<&,Xi,X2,s)= E /(Yg,*.Xi,X2.*). yePF\GF again convergent for &a;(s) > 1, can be rewritten as £(g,5) = Xl(detg)|detg|7 £ $[£rg]M2W(0 <**'• The Poisson summation formula now implies that E(g,s) has a meromorphic continuation to all 5, satisfies the functional equation E(g,*,xi,xi,') = £(g\*,xrl,x;M *) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 6 H. JACQUET AND D. ZAGIER (see "Notations and Conventions"), and is holomorphic unless Xi = X2> when it has simple poles at s = 0 and 5 = 1 with residues given by Res^1£(g,5) = ^(O)xi(detg), Resi=0£(g,5) = -i*(0)Xi(detg). The relationship between the two types of Eisenstein series is given by the following lemma. Lemma. /// e H(xv X2) " K-finite, then the section Sf(g,s) can be written as a finite linear combination Sf(g,s) = L{2s, XiXzVE PMf(g,*„ Xi, X2, *) I and correspondingly the Eisenstein series Ef(g, s) as Ef(g,s) = L(2s,xlX2YlLPi(s)E(g,t>l,XuX2^), i where O, e S0(/\2) and each P[{s) is the reciprocal of a polynomial in s and in q~s for finitely many places v which has no zeroes in the half-plane 0te(s) > 0. Proof. The space #(xi>X2) is tne restricted tensor product over all places v of analogously defined spaces Hv(x\v, X2i>)> and wc may assume that /= Y\vfv with /„ e Hv{xi„, X2v)< fv 1 ^or almost all finite v. We claim that for such an / we can write (with the obvious notations) P (5) (*) SJ0(gv,s) = , "-■^\fv(gv>®c>Xlv>X2»>S) Ev\2-S, X\vX2v) where <£„ is a Schwartz-Bruhat function on F2, equal to the characteristic function of R2V for almost all v, and Pv(s) is an elementary function of 5, equal to 1 for almost all v. From this it will follow that f(g,s) equals /)(5)L(25,x1X21)_1/(g^,Xi,X2^) withP = n Pv and *0.Xi0,X2B.*) = Xi,,(det*)/ A(0,t)k]xioX2l(t)\t\2vdxt and therefore (*) holds with Pv(s) = Lv(2s,XivX2~l), which is the reciprocal of a polynomial in q~s having no zeroes in <%e(s) > 0. If v is a real place, then (since / is AT-finite) we may assume that fv has the form x I cosd sin0\ _ i„e /u I -sin f? costfj License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use EISENSTEIN SERIES AND THE SELBERG FORMULA. II 7 for some n e Z such that Xi0X2u(-l) = (-1)"; then we set *,(*, y) = e-^2+y\y dx)w (x, y e F0) with c = sign(«) and find '.[(31 Z'»\*-^-A-'ML •~*i'f**fiw» - 0. We may assume that / has the form with p,q,r,u ^ Z and # + «=/> + /■ +a. Then we take 0„(x, v) = JC»S'y 0, so we can set P„(s) = Q(s)'1 as before. This completes the proof of the lemma. Notation. For our standard basis elements fa (a e A(xi, X2)) we will sometimes use the notations Pai, $„, (i G Ia) for the polynomials and Schwartz-Bruhat functions occurring in the decomposition of Sa(g, s) given by the lemma. 1.2. Whittaker functions. We now discuss the Fourier coefficients of the Eisenstein series. The Bruhat decomposition gives for ^(5) > 1 Ef(g,s) = Sf(g,s) + £ Sf{wvg,s) v^NF