GONZÁLEZ, CHUNG HANG KWAN, STEVEN J. MILLER, ROGER VAN PESKI, AND TIAN AN WONG

Transcription

1 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn AND BEYOND ENDOSCOPY arxiv: v [math.nt] 3 Sep 07 OSCAR E. GONZÁLEZ, CHUNG HANG KWAN, STEVEN J. MILLER, ROGER VAN PESKI, AND TIAN AN WONG Abstract. Recent work of Altuğ continues the preliminary analysis of Langlands Beyond Endoscopy proposal for GL by removing the contribution of the trivial representation by a Poisson summation formula. We show that Altuğ s method of smoothing real elliptic orbital integrals by an approximate functional equation extends to GLn. We also discuss the case of an arbitrary reductive group, and remaining obstructions for applying Poisson summation. Contents. Introduction. Preliminaries 4 3. The approximate functional equation 7 4. Smoothing of the real orbital integral 0 5. Remarks on general reductive groups 3 6. Applications and obstructions 4 References 6. Introduction.. Overview of Beyond Endoscopy. One of the key conjectures of the Langlands Program is the Functoriality Conjecture: given two reductive groups G and G, and an L-homomorphism of the associated L-group L G to L G, one expects a transfer of automorphic forms on G to automorphic forms on G. Most cases of functoriality known today fall under the banner of endoscopy, that is, where G is an endoscopic group of G. The problem of endoscopy is addressed by the stable trace formula, recently made unconditional by Ngô s solution of the Fundamental Lemma [Ngô]. Anticipating this, Langlands proposed a new strategy to attack the general case, referred to as Beyond Endoscopy [Lan]. If an automorphic form π on G is a functorial transfer from a smaller G, then one expects the L-function Ls,π,r to have a pole at s = for some representation r of L G. In particular, the order of Ls,π,r at s =, which we denote by m r π, should be nonzero if and only if π is a transfer. Langlands idea then is to weight the spectral terms in the stable trace formula by m r π, resulting in a trace formula whose spectral side detects only π for which the m r π is nonzero. Date: September 4, Mathematics Subject Classification. F7 and F66. Key words and phrases. Beyond Endoscopy, orbital integrals, approximate functional equation.

2 O.E. GONZÁLEZ, C.H. KWAN, S.J. MILLER, R. VAN PESKI, AND T.A. WONG Since in general Ls,π,r is not a priori defined at s =, we account for the weight factor by taking the residue at s = of the logarithmic derivative of Ls,π,r. This should lead to an r-trace formula, Scuspf r = lim logq v S N V N cuspf v, r. v V N wheres cusp representsthe usual stable trace formula, V N is a finite set ofvaluations of the global field F, and q v the order of the residue field of F v which is less than N see [Art, ] for details. Following Arthur [Art], the stable distribution should have a decomposition S r cusp f = G ιr,g ˆP G cusp f,. G where ˆP cuspf are called primitive stable distributions on elliptic beyond endoscopic groups G, and by primitive one means the spectral contribution to the stable trace formula of tempered, cuspidal automorphic representations that are not functorial images from some smaller group. These primitive distributions, giving a new primitive trace formula, are to be defined inductively, and one hopes to establish these from the r-trace formula. As is usual with trace formulae, one would like both the r-trace formula and the primitive trace formula to be an identity of spectral and geometric sides. But since one only wants tempered automorphic representations to contribute to., one has to first remove the contribution of the nontempered representations. Inspired by work of Ngô, a suggestion was put forth in [FLN] to apply Poisson summation to the elliptic contribution, over a linear space called the Steinberg-Hitchin base. Assuming the existence of a such a summation, they show that the dominant term should cancel with the contribution of the trivial representation, which can be viewed as the most nontempered representation. In related work, the recent thesis of Altuğ completes the preliminary analysis carried out in [Lan] for GL and the standard representation [Alt], see also [Alt]. Working over Q, and restricting ramification to the infinite prime, Altuğ applies a modified form of the Poisson summation described in [FLN] by expressing the volume factors as values of Hecke L-functions and a strategic application of the approximate functional equation, the singularities of the archimedean orbital integral can be overcome, which is necessary in order to apply Poisson summation. By a detailed analysis, Altuğ shows that not only does the trivial representation contribute to the dominant term of the dual sum, but also the continuous spectral term associated to the nontrivial Weyl element of GL. Based on this, Arthur outlinesin [Art] a list ofproblemsto be addressedin orderto establishthe primitive trace formula... Main result. In this paper, we study to what extent Altuğ s method generalizes to a more general reductive group, focusing on Problem III discussed in [Art]. In particular, we show that Altuğ s use of the approximate functional equation to smooth the singularities of real orbital integrals can be generalized to GLn. In Another possibility is to take the residue of Ls,π,r itself at s =, in which case mrπ is more complicated.

3 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn 3 particular, we apply Altuğ s method [Alt] to study the elliptic part of the trace formula of G = GLn, measγ fg γgdg,.3 G γa\ga γ ell where the sum is taken over elliptic conjugacy classes of GQ. See Section for precise definitions. Choosing test functions as in., we rewrite it as s γ L,σ Eθ ± γ Orbf q ;γ,.4 q ±p k trγ,...,trγ n where E is the extension of Q defined by the elliptic element γ, σ E the Galois representation appearing in the factorization ζ E s = ζ Q sls,σ E, and the product is taken over all primes q of Q. Here the L-value represents the global volume term measγ seen before. We show that the approximate functional equation can be used again to smooth the singularities of the archimedean orbital integrals, generalizing[alt, Proposition 4.]. The main result is the following. Theorem.. Let θ ± γ be defined as in.9, φ any Schwartz function on R, and α > 0. Then the function defined by is smooth. fx,...,x n = θ ± x,...,x n φ Dx,...,x n α.5 In particular, we will take φ to be the cutoff functions V s and V s in the approximate functional equation for L,σ E, so that the product L,σ E θ ± γ.6 is such that the singularities of the archimedean orbital integral, which lie along the vanishing set of the discriminant of γ, can be controlled. For the cases GLn with n > 3, we require Artin s conjecture as a simplifying assumption. This result represents a first step towards establishing. for higher rank groups. In particular, in order to apply Poisson summation over the Steinberg- Hitchin base, the singularities of the real orbital integrals must be addressed. Regarding this point, the method of the approximate functional equation has been the most successful up to now. While our application of the approximate functional equation is not completely analogous to Altuğ s see Section 6 for more detailed discussion, we expect that the techniques and considerations of this paper will be valuable to further attempts to generalize Altuğ s method..3. Outline. This paper is organized as follows. In Section, we introduce the necessary definitions and notation, and using the class number formula arrive at.4. In Section 3, we introduce the approximate functional equation for Artin L- functions. In Section 4, we describe the characterization of orbital integrals on real reductive groups, and prove Theorem.. Finally, in Section 5 we give indications on how our analysis can be generalized to general reductive groups, using work of Ono and Shyr on Tamagawa numbers of algebraic tori. In Section 6, we explain how our theorem fits into the framework of generalizing Altuğ s analysis to GLn, and the remaining obstructions for applying Poisson summation.

4 4 O.E. GONZÁLEZ, C.H. KWAN, S.J. MILLER, R. VAN PESKI, AND T.A. WONG.4. Acknowledgments. The authors were supported by NSF Grant DMS and Williams College; the third named author was additionally supported by NSF grants DMS65673 and DMS We thank James Arthur, Jasmin Matz, and Tasho Kaletha for helpful discussions, and S. Ali Altuğ and Bill Casselman for comments on preliminary version of this paper. We also thank the referee for a careful reading of the paper and helpful suggestions.. Preliminaries.. Notation. We follow closely the setting of [Lan] and [Alt]. Let G = GLn and A = A Q be the ring of adeles of Q. Denote by v any valuation of Q, q any finite prime, and p a fixed prime. An element γ GQ is said to be elliptic over Q if its characteristic polynomial is irreducible over Q. Throughout this paper, the word elliptic will be reserved for elliptic over Q. Let Z + be the set of all matrices in the center of GR with positive entries, and G γ be the centralizer of γ in G. The discriminant of γ is given by D γ = i<jγ i γ j,. where the γ i s are the distinct eigenvalues of γ. An elliptic element γ defines, by its characteristic polynomial, a degree n extension E of Q, such that for some integer s γ, and D E is the discriminant of E. D γ = s γ D E.. Definition.. Now let f = v f v be a function in C c GA. Define the global volume term measγ = measz + G γ Q\G γ A.3 and the orbital integral Orbf v ;γ = G γq v\gq v f v g γgdg v..4 Then the elliptic part of the Arthur-Selberg trace formula refers to measγ G γa\gafg γgdg = γ ellmeasγ Orbf v ;γ,.5 v γ ell where the sum is understood to be over representatives γ of elliptic conjugacy classes in GQ. Note that since G = GLn, these orbital integrals are in fact stable distributions. Definition.. Fix a prime p and integer k. The local test functions f v Cc Q v are chosen as follows: at the archimedean place, choose f Cc Z + \GR such that its orbital integrals are compactly supported; at finite places, choose f q and fp k such that f is supported on the set γ GZ with detγ = p k. Finally, we define f p,k by f p,k := f f p,k p q p f p,k q..6

5 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn 5 Remark.3. Note however, that this condition on the determinant is not necessary for our analysis, we only impose it to reflect more closely the setting of [Lan, ] and [Alt], where f p is chosen so that trπf p = trsym k Aπ p, where Aπ p is the Satake parameter of π at p, and the rest meaning we only allow ramification at infinity... Class number formula and measures. Denote by A E the adele ring of E, and I E = A E the ideles of E. Let v be the normalized absolute value on the completion E v and AE : I E R be the absolute value defined by x AE = v x v v,.7 where x = x v. Here. AE is a group homomorphism and we define the norm-one idele group to be its kernel, denoted IE, and E \IE the norm-one idele class group. Remark.4. The measures and test functions in.3 and.4 are to be chosen analogously as in [Alt]. We mention the choices once again here. We first describe the choice of measure on G. At any finite prime q, we choose any Haar measure on GQ q giving measure on GZ q, and the same with G γ Q q ; at infinity, we choose any Haar measure on GR. In keeping with [Alt, p.797], we note that there are more natural ways to normalize measures, but we make this choice so as to remain consistent with the hypotheses of [Lan, Alt]. Let γ be an elliptic element in GQ. Recall that it defines a degree n extension E of Q. It was observed by Langlands [Lan, Equation 9] that Z + G γ Q\G γ A = Z + E \I E = E \I E..8 Choosing the measures on Z + G γ Q\G γ A as such, we also require the measure on E \IE to be such that measz + G γ Q\G γ A = mease \IE..9 so that we have by [T, Theorem 4.3.], the relation mease \I E = r π r h E R E w E,.0 where h E is the class number of E, R E is the regulator of E, r is the number of real embeddings of E and r is the number of pairs of complex embeddings of E, and w E is the number of roots of unity in E. For a number field E, the Dedekind zeta function of E is defined by ζ E s = a N E/Q a s = p N E/Q p s. for Rs >, where the sum is over all non-zero integral ideals of E and the product is over all prime ideals of E. Hecke showed that the Dedekind zeta function ζ E s can be analytically continued to the whole complex plane except having a simple pole at s =. Recall the classical Dirichlet class number formula Res s= ζ E s = r π r h E R E,. w E DE which will be related to the idele class group via.0.

6 6 O.E. GONZÁLEZ, C.H. KWAN, S.J. MILLER, R. VAN PESKI, AND T.A. WONG Here again E is a number field. Let ρ : GalE/Q GLd,C be a finite dimensional Galois representation. The Artin L-function associated to ρ is given in terms of the Euler product: Ls,ρ = p deti ρfr p p s,.3 where Fr p is the Frobenius element in GalE/Q. In general, E/Q is not necessarily an Galois extension. In this case, the Artin L-function is obtained in the following manner: let E Gal be the Galois closure of E over Q, G = GalE Gal /Q, H = GalE Gal /E, and ρ G/H be the permutation representation of G on G/H. Then ζ E s = Ls,ρ G/H. The representation can in turn be decomposed into irreducible representations ρ i s on G, i.e., ρ G/H = n i ρ i, n i are integers, and we have the factorization ζ E s = i Ls,ρ i ni..4 Note that ζ E s has a simple pole at s =. We can also write ζ E s = ζ Q sls,σ E for some representation σ E. For more discussions on Artin L functions, we refer the reader to [IK, Mur, Neu]. By equations.0 and., we have and also mease \I E = D E L,σ E,.5 measz + G γ Q\G γ A = D E L,σ E..6 This is the form of the class number formula we shall use..3. Rewriting the elliptic term. By the choice of test function f, the right hand side of.5 is non-zero if and only if detγ q = for any finite prime q p and detγ p = p k. Therefore detγ Z q for any finite prime q and detγ Z. Moreover, detγ = p k. We parametrize γ GQ by the coefficients of its characteristic polynomial, where By.8 and detγ = ±p k, we write X n a X n + + n a n.7 a,...,a n = trγ,...,trγ n,detγ..8 θ ± γ = θ ± a,...,a n,±p k,.9 where θ γ ± = D γ / Orbf ;γ, since θ is invariant under conjugation, therefore we can consider θ ± γ as a function on Rn. The right hand side of.5 then becomes s γ L,σ Eθ ± γ q ±p k Orbf p,k q ;γ,.0 where the inner sum is taken over those a,...,a n over Z n corresponding to elliptic elements.

7 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn 7 Remark.5. The discriminant that appears in the class number formula is that of the number field, whereas the discriminant that appears in the orbital integral is that of the characteristic polynomial. We will see from the proof of Theorem 4. that the factor s γ does not play a significant role in the issue of smoothness. 3. The approximate functional equation In this section, we introduce the approximate functional equation of Artin L- functions, and show that the cutoff functions V s and V s in it are smooth and have good decay. For convenience, we assume the Artin conjecture, i.e., any Artin L-function attached to an irreducible, non-trivial and Galois representation of a number field is entire. Without the assumption, the following theorem is still valid, except that there would be additional terms accounting for possible contributions of poles of Ls,ρ along the lines Rs = 0,, which can be explicitly given. Theorem 3.. Let Ls, ρ be the Artin L-function associated to an irreducible, non-trivial, Galois representation ρ with conductor q = qρ, and assume the Artin Conjecture. Let Gu be any function which is holomorphic and bounded in the strip 4 < Ru < 4, even, and normalized by G0 =. Let X > 0 be a parameter to be chosen. Then for s in the strip 0 Rs, we have where Ls,ρ = n λ ρ n n n s V s X +ǫs,ρ q n λ ρ n Ls,ρ = n s, ǫs,ρ = ǫρq sγ s,ρ ; γs, ρ n= V s y = y u Gu γs+u,ρ du πi γs, ρ u 3 λ ρ n nx n s V s q, 3. and ǫρ is the root number of Ls,ρ and is a complex number with modulus. Proof. See Theorem 5.3 of [IK]. 3.. Gamma factors of Artin L-functions. The gamma factor of the Artin L-function is a product of local gamma factors γ v s,ρ over infinite places v of a number field E. Let r and r be as before, so that r + r = n, where n is the degree of the number field E. Let σ v be the Frobenius conjugacy class associated to the completion E v. Then σ v is of order if v is a real place, and trivial if v is complex. Hence, we have π ds/ Γ s γ v s,ρ = π ds/ Γ s dγ s+ d if v is a complex place d + v Γ s+ d v if v is a real place, 3. where d = degρ is the dimension of ρ and d + v, d v are the multiplicities, respectively, of the eigenvalue +, for ρσ v.

8 8 O.E. GONZÁLEZ, C.H. KWAN, S.J. MILLER, R. VAN PESKI, AND T.A. WONG Denote by d +,d the sum over real v of d + v and d v, respectively. The global gamma factor of the Artin L-function is the following: γs,ρ = v γ v s,ρ s d + s+ = π dsr/ Γ Γ d s d + = π dsr+r/ +dr s+ Γ Γ s dr π dsr/ dr s+ Γ Γ d +dr. 3.3 As before, E is the degree n number field defined by an elliptic element in GL n Q. Wehavethefactorizationζ E s=ζ Q sls,σ E inwhichthedegreeofthe representationσ E is n. If σ E is a reducible representation, then Ls,σ E further factorizes into a product of L-functions of non-trivial irreducible representations of GalE/Q. If n =, we have ζ E s = ζ Q sls,χ for some non-trivial quadratic Dirichlet character χ, where ζ Q is the Riemann zeta function. The further analysis is completed in [Alt]. It is a classical result, due to Brauer, stating that if E/Q is an abelian extension, then ζ E s is indeed a product of Dirichlet L-functions attached to distinct primitive characters and exactly one of the character is trivial. Now suppose n = 3 and E is a cubic field. Then the degree of the Galois representation σ E is. Either r,r =, or r,r = 3,0. We consider both cases as follow. Case : r,r =,. The cubic extension E/Q is not Galois, and σ E is an irreducible representation. Case : r,r = 3,0. The cubic extension is Galois, and σ E is a reducible representation, decomposing as χ and χ, where χ is the cubic character. The conjugacy class is order two and our character is order three, so we only get d + = and d = 0. Hence, we have the factorization: ζ E s = ζ Q sls,χls,χ, where Ls,χ is the Hecke L-function associated to the cubic character χ defined by E. In fact, Ls,χLs,χ can be written as the following Euler product χp= p s χp +p s +p s. One knows that the irreducible two-dimensional representation of GalE/Q = S 3 corresponds to a modular form, hence Ls,σ E is entire in Case. Case is simpler, and is known by class field theory. For n > 3, there could be Artin L-functions associated to irreducible representations of degree greater than in which assuming the Artin conjecture is necessary. Having an explicit expression of the Euler product.3 just like the one above would be a crucial step for the combination of the volume term with the q-adic orbital integrals in order to form a modified L-function, such as in [Alt, p.7], that would be a weighted sum of L-functions associated to the quadratic residue symbol. 3.. The cutoff functions. We now examine the cutoff functions in the approximate functional equation of Artin L functions for general Galois representation ρ with degree d. From equation.0, we only need the approximate functional

9 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn 9 equation for the case s =. Hence in the following we restrict s to be a real number in the range [0,]. Theorem 3.. Let s be a real number in the range [0,]. Then the functions V s and V s are smooth functions with the decay property that for any real number m > 3, as y, we have V s y,v s y y m. 3.4 Proof. ItisobviousthatV s isasmoothfunctionbydifferentiationundertheintegral sign. Next we consider its decay. The choice of the bounded holomorphic function G in the approximate functional equation will not be important to us, hence we will simply assume that Gu is identically. Then V s y = πiγs, ρ 3 y uγ s+u aγ s+u+ b du, 3.5 uπ ds+ur+r/ where a := d + +dr and b := d +dr. We will need the following form of Stirling s approximation. Let u = σ + it, < σ σ σ < + and t. Then we have Γu = π / t σ e π t / +O t. 3.6 uniformly in < σ σ σ < + as t. We would like to do a contour shift from 3 to m by considering the rectangle of height T that is symmetric about the x-axis with vertical sides at 3 and m. First we treat the integral along the top part of the contour [3+iT,m+iT]. We remark that the implicit constants in the estimations below may depend on s and ρ. The calculations for the bottom contour follows similarly and therefore will be omitted. First, we have m+it 3+iT m 3 y uγ s+u y σ Γ s+σ aγ s+u+ b du uπ ds+ur+r/ Then applying Stirling, this is m 3 and thus y σ π T +i T a Γ s+σ + +i T s+σ T a+bs a e a+bπt 4 e πt/4 a π T m 3 b π ds+σr +r / dσ σ +T. s+σ T a+b/ a+b/ yπ dr+r/ σdσ e πt/4 b π dσr +r / σ +T dσ, T a+bm+b a+bπt y m logt e Next we treat the integral along the right vertical contour [m it,m+it]. We first write m+it y uγ s+u aγ s+u+ b T m it Γs+m+it a Γ s+m+it+ b du = y m it uπ ds+ur+r/ T m+itπ ds+m+itr+r/idt. 3.8

10 0 O.E. GONZÁLEZ, C.H. KWAN, S.J. MILLER, R. VAN PESKI, AND T.A. WONG Then take absolute values and apply again Stirling s formula, we have y m T T Γ s+m +i t a Γ s+m+ +i t b dt m +t y m +y m T y m + c T c s+m t π e πt/4 s+m a t π e πt/4 b dt m +t t s+ma+b/ a/ e πta+b/4 dt 3.9 for some constant c > 0 large enough. We note that the integrand defining V s has no poles in the region Ru > 3 since Γz is holomorphic on Rz > 0. Therefore by Cauchy s Theorem, we have 3+iT 3 it y m + y uγ s+u aγ s+u+ b du uπ ds+ur+r/ T c t s+ma+b/ a/ e πta+b/4 dt + T a+bm+b y m logt e a+bπt 4. It is clear that the integral converges. Therefore, letting T, we have c 3 t s+ma+b/ a/ e πta+b/4 dt 3.0 y uγ s+u aγ s++u b du uπ ds+ur+r/ y m 3. and the result follows. 4. Smoothing of the real orbital integral We follow the exposition of [Alt, ] and substitute the approximate functional equation of Artin L-function into equation.0. This illustrates that the techniques used in [Alt, Alt] to handle the singularities of the real orbital integral hold for GLn. Suppose σ E = ρ ρ t, where ρ,...,ρ t are non-trivial irreducible representations on GalE Gal /Q. Then by the absolute convergence of the series in the approximate functional equation, we may expand the product of Artin L-functions as follows L,σ E = B {,...,t} n,...,n t i B λ ρi n i n i V i ni X i qi i B ǫ,ρ i λ ρi n i V i 0 ni X i qi. 4.

11 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn 4.. The singularities. We write the elliptic contribution as ni ±p k B {,...,t} n,...,n t i B s γ θ± γ q λ ρi n i V i n i X i qi i B ǫ,ρ i λ ρi n i V i 0 ni X i qi Orbf p,k q ;γ. 4. Again by absolute convergence, we may switch the order of the summations, yielding the following expression i B ±p k B {,...,t} n,...,n t j B V i ni X i qi n j s γ V i 0 i B ρi n i i Bλ i B ni X i θ ± qi γ q ǫ,ρ i λ ρi n i Orbfq p,k ;γ 4.3 and we would like to focus on the innermost sum. For i =,...,t, plug-in X i = D E Gal c with 0 < c < / and the expressions of the conductors q i c.f. pp.4 of [IK] are given by q i = D E Gal di N E Gal /Qf i, 4.4 with d i being the degree of ρ i and f i being the Artin conductor, which is a non-zero integral ideal of E Gal. Referring the discussion preceding Section 3, we now turn to the smoothness of the functions and V i V i 0 D E Gal d i +c D E Gal d i c n i θ γ, ± 4.5 N E /Qf Gal i n i N E Gal /Qf i θ ± γ. 4.6 Since D γ, D E Gal differ only by a non-zero integral multiple and N E Gal /Qf i are positive integers, the smoothing of the expressions 4.5 and 4.6 now follow as an immediate application of Theorem 4.. On the other hand, the behaviour of V 0 and V are discussed in the previous section, so it remains to understand the singularities of θ ± γ. Consider the real orbital integral Orbf ;γ. Let γ be a regular element in G, which is an n by n matrix with distinct eigenvalues, and T reg be the set of all regular elements in T = G γ. Also let f C c Z + \GR. Then it is well known that θ ± γ = D γ / Orbf ;γ 4.7 extends smoothly to T reg. This follows from the proof of Theorem 7. of [HC, p.45], and in particular Lemma 7.4 of [HC, p.47] and Lemma 40 of [HC, p.49]. Our function θ ± γ is related to the transform F T f of Harish-Chandra by a function of γ, which does not affect our analysis. Notice that z T reg if and only if D γ is nonzero. In other words, Orbf ;γ may have singularities on T \ T reg. Depending on the root, one either has a removable singularity or the jump relations of Harish-Chandra along the walls of T,

12 O.E. GONZÁLEZ, C.H. KWAN, S.J. MILLER, R. VAN PESKI, AND T.A. WONG though we do not need their precise form here c.f. [Sh, Sh, Bou]. Indeed, given fxg/dx, with gx-schwarz class, as long as the singularities of the function fx is on the zero locus of Dx and are not worse than x β for some β 0, then the product function is smooth. Note also that for p-adic orbital integrals, a similar behavior may also occur, but again, we do not consider them in this paper. 4.. Smoothing. The main theorem follows now from the preceding discussion. Theorem 4.. Let φ be any Schwartz function on R and α be a positive constant. Then the function defined by is smooth. fx,...,x n = θ x,...,x n φ Dx,...,x n α 4.8 Proof. By the discussions precede Theorem 4. and the fact that φ is a Schwartz function, we have f being a smooth function on T reg. Also, the singularities of the real orbital integral must lie on the zero locus of Dx,...,x n and they are not worse than x β for some β 0 β may depend on the choice of point of singularity. Now fix a singular point a = a,...,a n T reg. As x = x,...,x n a = a,...,a n, we have Dx α and φ Dx α M Dx Mα 4.9 for any positive constant M. Then choose any M > β +/α, we have fx x a β Dx Mα x a β Dx Qx a + Qx a Mα x a β Dx Qx a Mα + Q Mα x a Mα β = Dx Qx a Mα β Dx Da Qx a x a + Q Mα x a Mα β, β 4.0 where Q is the Jacobi matrix of D at x = a and Q denotes the matrix -norm of Q. By the continuity and differentiability of D, we have the last expression tends to 0 as x a. Therefore, lim fx = 0 4. x a and we can redefine fa = 0. Let h 0. By mean value theorem, we have fa +h,...,a n fa,...,a n h h β Da +h,...,a n Mα = h Mα β D a +θh,...,a n x for some θ,. By the continuity of D x and Mα β > 0, Mα 4. We thank S.A. Altuğ for pointing this out to us.

13 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn 3 f fa +h,...,a n fa,...,a n a,...,a n = lim = x h 0 h f Similarly, x i a,...,a n = 0 for i n. By carrying out the above argument inductively, we have all of the partial derivatives at x = a exists. Similar argument holds for the points on T \T reg which are not the singularities of the real orbital integral. Therefore, f is a smooth function on R n and this completes the proof. 5. Remarks on general reductive groups Now let G be a reductive group over Q. In this section we indicate how the preceding analysis can be extended to general G, though for general G we only consider unstable elliptic orbital integrals. As before, γ will be an elliptic element of GQ, so that G γ Q is a torus. Based on work of Ono [Ono], Shyr deduced a class number relation for tori. We briefly describe this, and refer to [Shy, 3] for details. 5.. Consider T = G γ as an algebraic torus over Q, and let ˆT = HomT,G m be the Z-module of rational characters of T. The torus T splits over a finite separable extension K of Q, and Γ = GalK/Q acts on ˆT. Then ˆT becomes a free Γ-module with rank r = dimt, and denote by χ T the character of the Γ-module. The character decomposes into h χ T = m i χ i 5. i=0 for some integer h. Here χ i are irreducible characters of G with χ 0 the principal character, and m i the multiplicity, whereby m 0 = r. It follows then that the Artin L-function factorizes as h Ls,χ T = ζs r Ls,χ i mi. 5. Moreover, for i, L,χ i is nonzero, so that the value h ρ T := lims r Ls,χ T = L,χ i 5.3 s is finite and nonzero, and is called the quasi-residue of T over Q. By [Ono], it is independent of choice of splitting field. Now, choosing canonical Haar measures related to the Tamagawa numbers, Shyr obtains the relation ρ T = h TR T i= τ T w T D / T i= 5.4 where τ T is the Tamagawa number of T, and the other h T,R T,w T, and D T are arithmetic invariants of T defined analogous to those appearing in Dirichlet s class number formula.. Then onemayproceed asin Section, and in particular,using.6 to writethe volume term as the value at of an Artin L-function, and apply the approximate functional equation. Then by similar estimates in Section 4 the singularities of the real orbital integral may be smoothed.

14 4 O.E. GONZÁLEZ, C.H. KWAN, S.J. MILLER, R. VAN PESKI, AND T.A. WONG Remark 5.. In the case of G = GLn the element γ is elliptic and defines a degree n extension E over Q, and the torus is simply the Weil restriction Res E/Q G m, split by K. By the remark following [Shy, Theorem ], one indeed recovers ρ T = Res s= ζ E s, 5.5 recovering the original case, and in particular the analytic class number formula. 6. Applications and obstructions In this section, we discuss how the main theorem generalizes Altuğ s analysis to GLn, and the briefly describe the remaining issues to be overcome in order to apply Poisson summation in this case: absorbing the p-adic orbital into the volume factor, and controlling the Fourier transforms in the dual sum. 6.. The p-adic orbital integrals. The main obstruction relates to the p-adic orbital integrals. By Equation 59 in [Lan,.5], the product of the p-adic orbital integrals for GL can be expressed using the Kronecker symbol, q G γq q\gq qf q g γgdḡ q = f s γ f q f D q q. 6. Then by [Alt,..], this can be combined with global volume factor by a change of variables to give D L, D f f s q f D q = q f s f L, m N/f, 6. where s D = m N and D a discriminant of a quadratic number field. Then, the sum of L-functions is itself shown to be an L-function, m L,m N = f L N/f, 6.3 where the sum is taken over those f satisfying f M N and m N/f 0, mod 4, and the approximate functional equation is applied to this final form. In order to generalize this to other groups, one would need an expression for the p-adic orbital integrals related to the Artin representation χ T as in Section 5. Unfortunately, for general groups a closed formula is not known for these integrals, though it is interesting to note that by [Ngô], and not to mention [Hal], one knows that evaluating such p-adic orbital integrals is closely related to counting points on varieties over finite fields. Remark 6.. In the case of GL3 the p-adic orbital integrals for the unit element of the Hecke algebra have in fact been computed explicitly by Kottwitz [Kot, p.66]. If the elliptic element γ = α + π n β generates an unramified cubic extension, one has p 3n+ p+p +p+ 3p n p +p++3 p, 6.4 p+ while for a ramified cubic extension one has p 3n+ p+p +valβ p n p +p+p valβ + p, 6.5 p+

15 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn 5 where valβ = or, but it does not seem clear what the analogous identity for 6. might be. 6.. Poisson summation. Supposing for the moment that we had such an expression like 6.3 for G = GLn, whereby the elliptic orbital integral measγ fg γgdg, which we expressed as can be simplified into f= ±p k γ ell l= G γa\ga s γ L,σ Eθ ± γ q ±p k m Z f m ±4p k Orbf q ;γ, 6.6 s γ L,σ E θ± γ 6.7 where σ E is another degree n Artin representation related to the original σ E. Then we could apply the approximate functional equation to L,σ E, and thus Theorem 4. to the resulting expression to allow for Poisson summation. Next, we formally complete the inner sum as in [Alt] by adding and subtracting the missing terms, the new subtracted terms either vanishing in the limit. or need to be treated separately. 3 In orderfor the Poissonsummation formula overthe inner sum on Z n to be valid, we must control the Fourier transform of the dual sum. For this, we would require a more precise characterization of the singularities of the orbital integrals than that discussed in Section 4.. Besides the main obstacles anticipated and addressed in [Alt, Alt] for applying the Poisson summation i.e., absolute convergence of L-series, summation over incomplete lattice and singularities contributed by the real orbital integral,there is one extra problem appearing in the higher rank cases which we would like to highlight: We first state the version of equation 4.3 in [Alt, Alt] with the missing terms correspond to D γ being perfect squares being added back in is: m ±4p k /f m θ f l l p k/ [ F lf m ±4p k α + lf m ±4p k H lf ] m ±4p k α, 6.8 where F and H are some Schwartz function, α 0,, and in the summation sign means we only sum over m such that m ±4p k /f 0, mod4. Not only are the coefficients of the L-series explicitly given by the Kronecker symbol, they are also periodic. If we want to apply Poisson summation to the m- sum, the summands have to be defined on R, smooth and with good decay. Of course, the Kronecker symbol is only defined for m Z. Altuğ crucially uses the periodicity to split the m-sum, bringing the Kronecker symbol out of the sum: 4 3 We thank the reviewer for pointing this out to us. 4 There is a slight mistake in [Alt, pp. 807]. After taking the Kronecker symbol out of the m- sum, it should no longer depend on m. It should be instead, as in equation 6.9. a ±4p k /f l

16 6 O.E. GONZÁLEZ, C.H. KWAN, S.J. MILLER, R. VAN PESKI, AND T.A. WONG = a mod 4lf m Z a ±4p k 0 mod f m a mod 4lf a ±4p k /f 0, mod 4 a mod 4lf a ±4p k 0 mod f a ±4p k /f 0, mod 4 a ±4p k /f l m ±4p k /f l θ m Z m a mod 4lf m θ G ± p k/ l,f m m p k/ G ± l,f m, 6.9 where G ± l,f m denotes the bracketed term in the m-sum of equation 6.8. But in generalforgln,n 3, thisonlyworksforelliptic γ defining anabelianextension, which is a particularly special case. References [Alt] S. A. Altuğ, Beyond Endoscopy via the Trace Formula - I Poisson Summation and Contributions of Special Representations, Composito Mathematica 5 05, [Alt] S. A. Altuğ, Beyond Endoscopy via the Trace Formula Ph. D. Thesis, available at [Art] J. Arthur, Problems Beyond Endoscopy, Roger Howe volume, preprint, available at [Bou] A. Bouaziz, Formule d inversion des intégrales orbitales sur les groupes de Lie réductifs, J. Funct. Anal , no., MR35994 [FLN] E. Frenkel, R. P. Langlands and B. C. Ngô, La formule des traces et la functorialité. Le debut dun Programme, Ann. Sci. Math. Québec 34 00, [Hal] T. Hales, Hyperelliptic curves and harmonic analysis why harmonic analysis on reductive p-adic groups is not elementary, Representation theory and analysis on homogeneous spaces New Brunswick, NJ, 993, 37 69, Contemp. Math. 77, Amer. Math. Soc., Providence, RI, 994. [HC] Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 9 975, [HC] Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc , [IK] H. Iwaniec and E. Kowalski, Analytic number theory, AMS Colloquium Publications, vol. 53, 004. [Kot] R. Kottwitz, Unstable orbital integrals on SL3, Duke Math. J , [Kot] R. Kottwitz, Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic analysis, the trace formula, and Shimura varieties, pp , Clay Math. Proc. 4, Amer. Math. Soc., Providence, RI, 005. [Lan] R. P. Langlands, Beyond Endoscopy, in Contributions to automorphic forms, geometry and number theory, pp , Johns Hopkins University Press, Baltimore, MD, 004. [Mur] Murty, M. Ram, Murty, V. Kumar, Non-vanishing of L-Functions and Applications, Birkhäuser Basel 997. [Neu] Neukirch, Jürgen, Algebraic Number Theory, Springer-Velag Berlin Heidelberg 999. [Ngô] B. C. Ngô, Le lemme fondamental pour les algebres de Lie, Publ. Math. Inst. Hautes Études Sci. 00, 69. available at [Ono] T. Ono, Arithmetic of algebraic tori, Ann. of Math , [Sh] D. Shelstad, Characters and inner forms of a quasi-split group over R, Compositio Math , no., 45. MR Also, for the same equation in [Alt, p. 796], on the summation sign should refer to summing over those f such that f m ±4p k and m ±4p k /f 0, mod 4 instead of f m ±4p k and m ±4p k /f k 0, mod4.

17 ON SMOOTHING SINGULARITIES OF ELLIPTIC ORBITAL INTEGRALS ON GLn 7 [Sh] D. Shelstad, Orbital integrals, endoscopic groups and L-indistinguishability for real groups, in Conference on automorphic theory Dijon, 98, 35 9, Publ. Math. Univ. Paris VII, 5, Univ. Paris VII, Paris. MR07384 [Shy] J. M. Shyr, On some class number relations of algebraic tori, Michigan Math. J , no. 3, [T] J. T. Tate, Fourier analysis in number fields, and Hecke s zeta-functions, in Algebraic Number Theory Proc. Instructional Conf., Brighton, 965, , Thompson, Washington, DC. address: oscar.gonzalez3@upr.edu Department of Mathematics, University of Puerto Rico, Río Piedras, PR address: kevinkwanch@gmail.com Department of Mathematics, the Chinese University of Hong Kong, Shatin, Hong Kong address: sjm@williams.edu, Steven.Miller.MC.96@aya.yale.edu Department of Mathematics and Statistics, Williams College, Williamstown, MA 067 address: rpeski@princeton.edu Department of Mathematics, Princeton University, Princeton, NJ address: t.wong@columbia.edu Max Planck Institut für Mathematik, Vivatsgasse 7, Bonn, 53