Adrian Diaconu Paul Garrett

Transcription

1 INTEGRAL MOMENTS OF AUTOMORPHIC L FUNCTIONS Adrian Diaconu Paul Garrett Abstract. This paper exposes the underlying mechanism for obtaining second integral moments of GL automorphic L functions over an arbitrary number field. Here, moments for GL are presented in a form enabling application of the structure of adele groups and their representation theory. To the best of our knowledge, this is the first formulation of integral moments in adele-group-theoretic terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers Q, we recover the classical results.. Introduction For ninety years, the study of mean values of families of automorphic L functions has played a central role in analytic number theory for their applications to classical problems. In the absence of the Riemann Hypothesis, or the Grand Riemann Hypothesis, rather, when referring to general L functions, suitable mean value results often served as a substitute. In particular, obtaining asymptotics or sharp bounds for integral moments of automorphic L functions is of considerable interest. The study of integral moments was initiated in 98 by Hardy and Littlewood (see [Ha-Li] who obtained the second moment of the Riemann zeta-function (. T ζ ( + it dt T log T About 8 years later, Ingham in [I] obtained the fourth moment (. T ζ ( + it 4 dt T (log T 4 π Subsequently, many papers by various authors were devoted to this subject. For instance see [At], [H-B], [G], [M], [J]. Most of the existing results concern integral moments of automorphic L functions defined over Q. No analog of (. or (. is known over an arbitrary number field. The only previously known results, for fields other than Q, are in [M4], [S], [BM], [BM] and [DG], all over quadratic fields. Here we expose the underlying mechanism to obtain second integral moments of GL automorphic L functions over an arbitrary number field. Integral moments for GL are presented in a form 99 Mathematics Subject Classification. R4, Secondary F66, F67, F7, M4, R47. Key words and phrases. Integral moments, Poincaré series, Eisenstein series, L functions, spectral decomposition, meromorphic continuation. Typeset by AMS-TEX

2 ADRIAN DIACONU PAUL GARRETT amenable to application of the representation theory of adele groups. To the best of our knowledge, this is the first formulation of integral moments on adele-groups, distinguishing global and local questions, and allowing uniform application to number fields. Precisely, for f an automorphic form on GL and χ an idele class character of the number field, let L(s, f χ denote the twisted L function attached to f. We obtain asymptotics for averages (.3 χ L ( + it, f χ M χ (t dt for suitable smooth weights M χ (t. The sum in (. is over a certain (in general infinite set of idele class characters. For general number fields, it seems that (. is the correct structure of the second integral moment of GL automorphic L functions. This was first pointed out by Sarnak in [S], where an average of the above type was studied over the Gaussian field Q(i; see also [DG]. From the analysis of Section, it will become apparent that this comes from the Fourier transform on the idele class group of the field. The course of the argument makes several points clear. First, the sum (over twists of moments of L functions is presented by an integral representation. Second, the kernel arise from a collection of local data, wound up into an automorphic form, and the proof proceeds by unwinding. Third, the local data at finite primes is of a mundane sort, already familiar from other constructions of Poincaré series. Fourth, the only subtlety resides in choices of archimedean data. Once this is understood, it is clear that Good s original idea in [G], seemingly limited to GL (Q, exhibits a good choice of local data for real primes. (See also [DG]. Similarly, while [DG] explicitly addresses only GL (Z[i], the discussion there does exhibit a good choice of local data for complex primes. That is, these two examples suffice to illustrate the (non-obvious choices of local data for all archimedean places. For applications, we will combine careful choices of archimedean data with extensions of the estimates in [Ho-Lo] and [S] (or [BR] to number fields. However, for now, we are content with a formulation suitable for both applications and extensions. In subsequent papers we will address convexity breaking in the t-aspect, and extend this approach to GL(n.. Unwinding to Euler product Let k be a number field, G = GL over k, and define the standard subgroups: {( } {( } {( } P = N = M = Z = center of G Also, for any place ν of k, let Kν max be the standard maximal compact subgroup. That is, for finite ν, we take Kν max = GL (o ν, at real places Kν max = O(, and at complex places Kν max = U(. We form a suitable Poincaré series, and unwind a corresponding global integral to express it as an inverse Mellin transform of an Euler product. The Poincaré series of the form (. Pé(g = ϕ(γg (g G A γ M k \G k for suitable function ϕ on G A. Let ϕ be a monomial vector ϕ = ν ϕ ν

3 INTEGRAL MOMENTS OF AUTOMORPHIC L FUNCTIONS 3 where for finite primes ν, the local component ϕ ν is defined as follows. For the usual maximal compact open subgroup Kν max of G ν, and for a character χ,ν of M ν, put (. ϕ ν (g = { χ,ν (m for g = mk, m M ν and k K max ν for g M ν K max ν (ν finite For ν infinite, for now we do not entirely specify ϕ ν, only requiring left equivariance (.3 ϕ ν (mnk = χ,ν (m ϕ ν (n (for ν infinite, m M ν, n N ν and k K max ν Require that χ = ν χ,ν be M k invariant. For clarity, and not needing anything more general, assume that v ( ( (.4 χ,ν (m = y y m = M y ν, v C y ν Thus, ϕ has trivial central character. Then ϕ is left M A equivariant by χ. For ν infinite, our assumptions imply that ( x x ϕ ν is a function of x only. Let f and f be cuspforms on G A. Eventually we will take f = f, but for now merely require the following. At all ν, require (without loss of generality that f and f have the same right K ν type, that this K ν type is irreducible, and that f and f correspond to the same vector in the K-type (up to scalar multiples. (Schur s lemma assures that this makes sense, insofar as there are no non-scalar automorphisms. Suppose that the representations of G A generated by f and f are irreducible, with the same central character. Last, require that each f i is a special vector locally everywhere in the representation it generates, in the following sense. Let (.5 f i (g = ξ Z k \M k W i (ξg be the Fourier expansion of f i, and let W i = ν W i,ν be the factorization of the Whittaker function W i into local data. By [JL], we may require that for all ν < the Hecke type local integrals ( y W i,ν y s ν dy y k ν differ by at most an exponential function from the local L factors for the representation generated by f i. Eventually we will take f = f, compatible with these requirements. The integral under consideration is (with notation suppressing details (.6 I(χ = Pé(g f (g f (g dg Z A G k \G A

4 4 ADRIAN DIACONU PAUL GARRETT For χ (and archimedean data in the range of absolute convergence, the integral unwinds (via the definition of the Poincaré series to Z A M k \G A ϕ(g f (g f (g dg Using the Fourier expansion this further unwinds to (.7 f (g = W (ξ g ξ Z k \M k ϕ(g W (g f (g dg Z A \G A Let C be the idele class group GL (A/GL (k, and Ĉ its dual. More explicitly, by Fujisaki s Lemma (see Weil [W], the idele class group C is a product of a copy of R + and a compact group C. By Pontryagin duality, Ĉ R Ĉ with Ĉ discrete. It is well-known that, for any compact open subgroup U fin of the finite-prime part in C, the dual of C /U fin is finitely generated with rank [k : Q]. The general Mellin transform and inversion are (.8 f(x = bc = χ b C C f(yχ(y dy χ (x dχ f(yχ (y y s dy χ (x x s ds πi R(s=σ C for suitable Haar measure on C. To formulate the main result of this section, we need one more piece of notation. For ν infinite and s C, let (.9 and K ν (s, χ,ν, χ ν = Z ν \M ν N ν Z ν \M ν ϕ ν (m ν n ν W,ν (m ν n ν W,ν (m νn ν χ ν (m ν m ν s ν (. K (s, χ, χ = ν K ν (s, χ,ν, χ ν χ ν (m ν m ν s ν dm ν dn ν dm ν Here χ = ν χ,ν is the character defining the monomial vector ϕ, and χ = ν χ ν Ĉ. Call the monomial vector ϕ admissible, if the series (. defining Pé(g converges absolutely and uniformly on compact sets, and furthermore, there exist distinct positive numbers a and b such that the integral defining K ν (s, χ,ν, χ ν converges absolutely for all ν, χ Ĉ and a < R(s < b. We do not need a complete determination of the class of admissible archimedean data, since we will use a limited range of explicit choices. For instance, take

5 INTEGRAL MOMENTS OF AUTOMORPHIC L FUNCTIONS 5 (. ϕ ν (g = ( w ( y v y y x for ν real, and g = G ν x +y ( w ( (v, w C y v y y x for ν complex, and g = G ν x + y Then the monomial vector ϕ generated by (.4 and (. is admissible for R(v and R(w sufficiently large. This choice of ϕ will be used in Section 4 to derive an asymptotic formula for the GL integral moment over the number field k. The main result of this section is Theorem.. For ϕ an admissible monomial vector as above, for suitable σ >, I(χ = L(χ χ s, f L(χ s, πi f K (s, χ, χ ds χ b C R(s=σ Let S be a finite set of places including archimedean places, all absolutely ramified primes, and all finite bad places for f and f. Then the sum is over a set Ĉ,S of characters unramified outside S, with bounded ramification at finite places, depending only upon f and f. Proof: Applying (.8 to f via the identification {( } y : y C C and using the Fourier expansion f (g = ξ Z k \M k W (ξ g the integral (.7 is = bc ( ϕ(g W (g f (m g χ(m dm dχ dg Z A \G A bc C ϕ(g W (g W (ξm g χ(m dm dg dχ Z A \G A C ξ Z k \M k ( = ϕ(g W (g W (m g χ(m dm dg dχ bc Z A \G A J where J is the ideles. For fixed f and f, the finite-prime ramification of the characters χ Ĉ is bounded, so there are only finitely many bad finite primes for all the χ which appear. In particular, all the characters χ which appear are unramified outside S and with bounded ramification, depending only on f and f, at finite places in S. Thus, for ν S finite, there exists a compact open subgroup U ν of o ν such that the kernel of the ν th component χ ν of χ contains U ν for all characters χ which appear.

6 6 ADRIAN DIACONU PAUL GARRETT Since f and f generate irreducibles locally everywhere, the Whittaker functions W i factor W i ({g ν : ν } = Π ν W i,ν (g ν Therefore, the inner integral over Z A \G A and J factors over primes, and I(χ = bc Π ν ( Z ν \G ν k ν ϕ ν (g ν W,ν (g ν W,ν (m νg ν χ ν (m ν dm ν dg ν dχ Let ω ν be the ν th component of the central character ω of f. Define a character of M ν by ( y y ( ( y y /y ω ν χ y ν Still denote this character by χ ν, without danger of confusion. In this notation, the last expression of I(χ is I(χ = bc Π ν ( Z ν \G ν Z ν \M ν ϕ ν (g ν W,ν (g ν W,ν (m νg ν χ ν (m ν dm ν dg ν Suppressing the index ν, the ν th local integral is ϕ(g W (g W (m g χ(m dm dg Z\G Take ν finite such that both f and f are right Kν max invariant. Use a ν adic Iwasawa decomposition g = mnk with m M, n N, and k Kν max. The Haar measure is d(mnk = dm dn dk with Haar measures on the factors. The integral becomes ϕ(mn W (mn W (m mn χ(m dm dn dm N To symmetrize the integral, replace m by m m to obtain ϕ(mn W (mn W (m n χ(m χ(m dm dn dm N The Whittaker functions W i have left N equivariance so and similarly for W. Thus, letting X(m, m = W i (ng = ψ(n W i (g (fixed non-trivial ψ W (mn = W (mnm m = ψ(mnm W (m N ϕ(n ψ(mnm ψ(m nm dn dχ

7 the local integral is INTEGRAL MOMENTS OF AUTOMORPHIC L FUNCTIONS 7 χ (m W (m W (m χ(m χ (m X(m, m dm dm We claim that for m and m in the supports of the Whittaker functions, the inner integral X(m, m is constant, independent of m, m. (And is for almost all finite primes. First, ϕ(mn is, unless n M K max N, that is, unless n N K max. On the other hand, ψ(mnm W (mk = ψ(mnm W (m = W (mn = W (m (for n N K Thus, for W (m, necessarily ψ(mnm =. A similar discussion applies to W. So (up to normalization the inner integral is for m, m in the supports of W and W. Then χ (m W (m W (m χ(m χ (m dm dm = (χ χ (m W (m dm χ(m W (m dm = L ν (χ,ν χ ν / ν, f L ν (χ ν / ν, f i.e., the product of local factors of the standard L functions in the theorem (up to exponential functions at finitely-many finite primes by our assumptions on f and f. For non-trivial right K type σ, the argument is similar but a little more complicated. The key point is that the inner integral over N (as above should not depend on mk and m k for mk and m k in the support of the Whittaker functions. Changing conventions for a moment, look at V σ valued Whittaker functions, and consider any W in the ν th Whittaker space for f i having right K isotype σ. Thus, W (gk = σ(k W (g (for g G and k K For ϕ(mn, again n N K. Then σ(k ψ(mnm W (m = W (mnk = σ(k W (mn = σ(k σ(n W (m where in the last expression n comes out on the right by the right σ equivariance of W. For m in the support of W, σ(n acts by the scalar ψ(mnm on W (mk, for all k K. Thus, σ(n is scalar on that copy of V σ. At the same time, this scalar is σ(n, so is independent of m if W (m. Thus, except for a common integral over K, the local integral falls into two pieces, each yielding the local factor of the L function. The common integral over K is a constant (from Schur orthogonality, non-zero since the two vectors are collinear in the K-type. At this point the archimedean local factors of the Euler product are not specified. The option to vary the choices is essential for applications. 3. Spectral decomposition of Poincaré series The objective now is to spectrally decompose the Poincaré series defined in (.. As we shall see, in general Pé(g is not square-integrable. However, choosing the archimedean part of the monomial vector ϕ to have enough decay, and after an obvious Eisenstein series is subtracted, the Poincaré

8 8 ADRIAN DIACONU PAUL GARRETT series is not only in L but also has sufficient decay so that its integrals against Eisenstein series converge absolutely. In particular, if the archimedean data is specialized to (., the Poincaré series Pé(g has meromorphic continuation in the variables v and w. This is achieved via spectral decomposition and meromorphic continuation of the spectral fragments, with estimates on the decomposition coefficients. See [DG], [DG] when k = Q, Q(i. Let k be a number field, G = GL over k, and ω a unitary character of Z k \Z A. Recall the decomposition The orthogonal complement L (Z A G k \G A, ω = L cusp(z A G k \G A, ω L cusp(z A G k \G A, ω L cusp(z A G k \G A, ω { dimensional representations} Ind G ν P ν (χ ν δν / dχ (GL (k\gl (Ab ν where δ the modular function on P A, and the isomorphism is via Eisenstein series. The projection to cuspforms is straightforward componentwise. We have Proposition 3.. Let f be a cuspform on G A generating a spherical representation locally everywhere, and suppose f corresponds to a spherical vector everywhere locally. In the region of absolute convergence of the Poincaré series Pé(g, the integral Z A G k \G A f(g Pé(g dg is an Euler product. At finite ν, the corresponding local factors are L ν (χ,ν / ν, f (up to a constant depending on the set of absolutely ramified primes in k. Proof: The computation uses the same facts as the Euler factorization in the previous section. Using the Fourier expansion f(g = W (ξg ξ Z k \M k unwind f(g Pé(g dg = Z A G k \G A Z A M k \G A W (ξg ϕ(g dg = W (g ϕ(g dg Z A \G A ξ ( = ν Z ν \G ν W ν (g ν ϕ ν (g ν dg ν At finite ν, suppressing the subscript ν, the integrand in the ν th local integral is right Kν max invariant, so we can integrate over MN with left Haar measure. The ν th Euler factor is W (mn ϕ(mn dn dm\ = ψ(mnm W (m χ (m ϕ(n dn dm N N

9 INTEGRAL MOMENTS OF AUTOMORPHIC L FUNCTIONS 9 for all finite primes ν. The integral over n is ψ(mnm ϕ(n dn N For ϕ(n to be non-zero requires n to lie in M K, which further requires, as before, that n N K. And, again, W (m = unless m(n Km N K The character ψ is trivial on N K. Thus, the integral over N is really the integral of over N K. Thus, at finite primes ν, the local factor is W (m χ (m dm = L ν (χ,ν / ν, f Of course, the spectral decomposition of a right K A invariant automorphic form only can involve everywhere locally spherical cuspforms. Assume that ϕ is given by (.. Taking R(v and R(w sufficiently large to ensure absolute convergence of Pé(g, the local integral in Proposition 3. at infinite ν is where, up to a constant, Z ν \G ν W ν (g ν ϕ ν (g ν dg ν = G ν ( + i µ f,ν ; v, w (3. G ν (s; v, w = π v Γ( ( v+ s Γ v+w s ( Γ v+s ( Γ v+w+s Γ ( ( w Γ v + w for ν real, and v Γ(v + sγ(v + w sγ(v + sγ(v + w + s (3.3 G ν (s; v, w = (π Γ(wΓ(v + w for ν complex. In the above expression µ f,ν denotes the local parameter of f at ν. Then, the sum corresponding to discrete spectrum in the spectral decomposition of Pé(g converges absolutely for (v, w C, apart from the poles of G ν ( + i µ f,ν ; v, w. For the remaining decomposition, subtract (as in [DG], [DG] a finite linear combination of Eisenstein series from the Poincaré series, leaving a function in L with sufficient decay to be integrated against Eisenstein series. The correct Eisenstein series to subtract becomes visible from the dominant part of the constant term of the Poincaré series (below. Write the Poincaré series as Pé(g = ϕ(γg = ϕ(βγg γ M k \G k β N k γ P k \G k By Poisson summation (3.4 Pé(g = γ P k \G k ψ (N k \N A b ϕ γg (ψ

10 ADRIAN DIACONU PAUL GARRETT where, ϕ g (n = ϕ(ng, and ϕ is Fourier transform along N A. The trivial ψ (that is, with ψ = Fourier term (3.5 ϕ γg ( γ P k \G k is an Eisenstein series, since the function g ϕ g ( = ϕ(ng dn N A is left M A equivariant by the character δχ, and left N A invariant. For ξ M k, ϕ ξg (ψ = ψ(n ϕ(nξg dn N A = ψ(n ϕ(ξ ξ nξ g dn = ψ(ξnξ ϕ(n g dn = ϕ g (ψ ξ N A N A where ψ ξ (n = ψ(ξnξ, by replacing n by ξnξ, using the left M k invariance of ϕ, and invoking the product formula to see that the change-of-measure is trivial. Since this action of Z k \M k is transitive on non-trivial characters on N k \N A, for a fixed choice of non-trivial character ψ, the sum over non-trivial characters can be rewritten as a more familiar sort of Poincaré series ϕ γg (ψ = ϕ γg (ψ ξ γ P k \G k ψ (N k \N A b γ P k \G k ξ Z k \M k = ϕ ξγg (ψ = ϕ γg (ψ ξ Z k \M k γ P k \G k γ Z k N k \G k Denote this version of the original Poincaré series, with the Eisenstein series subtracted, by (3.6 Pé (g = ϕ γg (ψ = Pé(g ϕ γg ( γ Z k N k \G k γ P k \G k Granting that Pé is not only in L (Z A G k \G A but also has sufficient decay so that its integrals against Eisenstein series (with parameter in a bounded vertical strip containing the critical line converge absolutely, the continuous-spectrum components of Pé are computed by integrals against Eisenstein series E(g = γ P k \G k η(γg for η left P k invariant, left M A equivariant, and left N A invariant. The Fourier expansion of this Eisenstein series is E(g = ψ (n E(ng dn N k \N A ψ (N k \N A b For a fixed non-trivial character ψ, the ψ th Fourier term is ψ(n E(ng dn = ψ(n N k \N A N k \N A γ P k \G k η(γng dn

11 INTEGRAL MOMENTS OF AUTOMORPHIC L FUNCTIONS = w P k \G k /N k (N k w P k w\n A ψ(n η(wng dn = ψ(n η(ng dn + N k \N A ψ(n η(w o ng dn N A ( = + ψ(n η(w o ng dn (where w o = N A because ψ is non-trivial and η is left N A invariant. Denote the ψ th Fourier term by (3.7 With parameters in a suitable range, = W (g = W η,ψ (g = ψ(n η(w o ng dn N A Pé (g E(g dg = Z A G k \G A Z A N A \G A Z A N A \G A N k \N A ϕ ng (ψ E(ng dn dg ϕ g (ψ ψ(ne(ng dn dg = ϕ g (ψ W (g dg N k \N A Z A N A \G A When η and ϕ are monomial in the respective (restricted tensor products of local representations, the latter integral is an Euler product. At finite primes ν, we evaluate the ν th Euler factor. In this case, the local components of both η and ϕ are right K ν invariant with maximal compact K ν. In Iwasawa coordinates nmk, with n N ν, m M ν and k K ν, a Haar measure is d(nmk = δ (m dn dm dk Thus, suppressing subscripts, the local integral is (3.8 ϕ m (ψ W (m δ (m dm There is no residual contribution to the spectral decomposition of Pé (g. Now we describe the continuous part of the spectral decomposition. At every place ν, let η ν be the spherical vector in the (non-normalized principal series Ind G ν P ν χ ν, normalized by η ν ( =. Take η = ν η ν. The corresponding Eisenstein series is E χ (g = γ P k \G k η(γg For any left Z A G k invariant and right K A invariant square-integrable f on G A, write f, E χ = f(g E χ (g dg Z A G k \G A

12 ADRIAN DIACONU PAUL GARRETT With suitable normalization of measures, continuous spectrum part of f = f, E χ E χ dχ R(χ= This formula requires isometric extensions to L of integral formulas that converge literally only on a smaller dense subspace (pseudo-eisenstein series. Happily, applying this to Pé, for R(χ sufficiently large in comparison to R(χ the integrals Pé, E χ were computed above in (3.8, yielding the same form of local integrals (at all finite places. Assume ϕ is defined by (.. Then the integrals Pé, E χ can be evaluated in terms of (3. and (3.3 at all infinite places. For fixed character χ, the corresponding integral of Eisenstein series can be meromorphically continued by shifting the vertical line as in [DG]. (The normalization of the Poincaré series in the latter reference is slightly different than here: in that paper the Poincaré series was normalized by needlessly multiplying by the gamma factor π w/ Γ(w/. The sum over the characters χ is absolutely convergent giving the meromorphic continuation of the Poincaré series Pé(g. If v =, the Poincaré series has, as a function of the variable w, a pole of order r + r + at w =, otherwise being holomorphic for R(w > c, with c > sufficiently small. As usual, r and r denote the number of real and complex embeddings of k, respectively. We record this discussion as Theorem 3.9. Assume ϕ is defined by (.. The Poincaré series Pé(g, originally defined for R(v and R(w large, has meromorphic continuation to a region in C containing v =, w =. As a function of w, for v =, it is holomorphic in the half-plane R(w > c for small positive constant c, except when w = where it has a pole of order r + r Asymptotic formula Let k be a number field with r real embeddings and r complex embeddings. Assume that ϕ is specialized to (.. By Theorem., for R(v and R(w sufficiently large, the integral I(χ = I(v, w defined by (.6 is (4. I(v, w = χ b C,S πi R(s=σ L(χ v+ s, f L(χ s, f K (s, v, w, χ ds where K (s, v, w, χ is given by (.9 and (., and where the sum is over χ Ĉ unramified outside S and with bounded ramification, depending only on f and f. By Theorem 3.9, it follows that I(v, w admits meromorphic continuation to a region in C containing the point v =, w =. In particular, if f = f = f, then for c > sufficiently small, the function I(, w is holomorphic for R(w > c, except for w =, where it has a pole of order r + r +. We will shift the line of integration to R(s = in (4. and set v =. To do so, we need some analytic continuation and good polynomial growth in I(s for the kernel function K (s, v, w, χ. In fact, it is desirable for applications to have precise asymptotic formulae as the parameters s, v, w, χ vary. By the decomposition (., the analysis of the kernel K (s, v, w, χ reduces to

13 INTEGRAL MOMENTS OF AUTOMORPHIC L FUNCTIONS 3 the corresponding analysis of the local component K ν (s, v, w, χ ν, for ν. When ν is complex, one can use the asymptotic formula already established in [DG], Theorem 6.. For coherence, we include a simple computation matching, as it should, the local integral (.9, for ν complex, with the integral (4.5 in [DG]. Fix a complex place ν. Recall that any character χ ν of Z ν \M ν C has the form χ ν (m ν = z ν lν +itν z l ν C ν ( m ν = ( zν, t ν R, l ν Z Assuming f and f have trivial K type, the local integral (.9 becomes π π K ν (s, v, w, χ ν = ( x + w e πi Tr C/R(y xe iθ y xe iθ Replacing x by x/y, we obtain K ν (s, v, w, χ ν = If we further substitute C π π y v+ s it ν K iµ (4πy y s+it ν K iµ (4πy e il νθ e il νθ dθ dθ dx dy dy C π π π π ( y x + y w e C/R( πi Tr xe iθ y y xe iθ y v s it ν K iµ (4πy y s+it ν K iµ (4πy e il νθ e il νθ dθ dθ dx dy dy y = r cos φ, x = r sin φ cos θ, x = r sin φ sin θ, y = u cos φ with φ π and θ π, then K ν (s, v, w, χ ν = π π π π π π (cos φ w+v e πi Tr C/R(r sin φ e i(θ+θ u sin φ e i(θ+θ r v+ s it ν K iµ (4πr cos φ u s+it ν K iµ (4πu cos φ e il νθ e il νθ sin φ dθ dθ dθ dφ dr du Using the Fourier expansion we obtain K ν (s, v, w, χ ν = (π 3 e it sin θ = π k= J k (t e ikθ K iµ (4πr cos φk iµ (4πu cos φj lν (4πr sin φj lν (4πu sin φ u s+it ν r v+ s it ν (cos φ w+v sin φ dφdrdu ru In the notation of [DG], equation (4.5, this is essentially K lν (s + it ν, v, w. It follows that K ν (s, v, w, χ ν is analytic in a region D : R(s = σ > ɛ, R(v > ɛ and R(w > 3 4, with a fixed (small ɛ >, and moreover, we have the asymptotic formula K ν (s, v, w, χ ν = π v+ A(v, w, µ, µ ( + l ν + 4(t + t ν w ( ( ] (4. [ + O σ, v, w, µ, µ + l ν + 4(t + t ν

14 4 ADRIAN DIACONU PAUL GARRETT where (4.3 A(v, w, µ, µ = w v 4 Γ ( w+v+iµ +iµ Γ ( w+v iµ +iµ Γ ( w+v+iµ iµ Γ (w + v ( Γ w+v iµ iµ For ν real, the corresponding argument is even simpler (see [DG] and [Zh]. In this case, the asymptotic formula of K ν (s, v, w, χ ν becomes (4.4 K ν (s, v, w, χ ν = B(v, w, µ, µ ( + t + t ν w [ ( ( + O σ, v, w, µ, µ + t + tν ] where B(v, w, µ, µ is a similar ratio of gamma functions. It now follows that for R(w sufficiently large, (4.5 I(, w = χ b C,S π L ( χ it, f L ( χ +it, f K ( + it,, w, χ dt In the above we assumed for simplicity that both local parameters µ = µ and µ f,ν = µ at ν f,ν are real. Since I(, w has analytic continuation to R(w > c, a mean value result can already be established by standard arguments. For instance, assume f = f = f, and choose a function h(w which is holomorphic and with sufficient decay (in I(w in a fixed vertical strip. One can choose a suitable product of gamma functions, for example. Consider the integral (4.6 i R(w=L I(, w h(w T w dw with L a large positive constant. Taking h( =, we have the asymptotic formula (4.7 χ b C,S L( + it, f χ M χ, T (t dt A T (log T r +r for some computable positive constant A, where (4.8 M χ, T (t = K ( i + it,, w, χ h(w T w dw For a character χ Ĉ, put κ χ (t = ν ν real R(w=L t + t ν ν ν complex ( l ν + 4(t + t ν (t R where t ν and l ν are the parameters of the local component χ ν of χ. Since χ is trivial on the positive reals, α ν t ν = ν with α ν = or according as ν is real or complex. Then, the main contribution to the asymptotic formula (4.7 comes from terms for which κ χ (t T.

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