Introduction
Trace Formula: from Arthur-Selberg to Jacquet-Zagier
Statement of the Main Results
Basic Notation
Some Applications
Idea of Proof and the Structure of the Thesis
Further Relevant Results
Contributions from Geometric Sides
Structure of G(F) -Conjugacy Classes
Then one has the decomposition (2.5) and both of the minimal polynomial and characteristic polynomial of AVij = A |F[A]αi,j are powers of ℘i(λ). Then by Lemma 17, for each such j, one can find an elementxj of the form in (2.7) such that the0jdj-row ofxj is exact.
Contributions from Nonsingular Conjugacy Classes
For any γ ∈G(F), write γP0(F) for the P0(F) conjugacy class of γ which is the same as the P(F) conjugacy class of γ. Then, by Corollary 32, one can decompose ZG(F)\G(F) )as. Therefore, the arguments in the proof of Theorem H (with V(s, λ) removed) work here for all Re(r)> 0. Then Corollary 69 follows from the functional.
Mirabolic Fourier Expansions of Automorphic Forms
Fix an integer ≥ 2. The maximal unipotent subgroup of G(AF), denoted by N(AF), is defined to be the set of all n×n upper triangular matrices in G(AF) with ones on the diagonal and arbitrary entries above diagonal. For 1 ≤ k ≤ n−1, let Bn−k be the standard Borel subgroup (ie, the subgroup consisting of nonsingular upper triangular matrices) of GLn−k; let Nn−k be the one.
Let S(π,Φ) be the finite set of non-Archimedean sites such that πv is unramified and Φv = Φ◦v is the characteristic function of G(OR,v)outside ΣF,∞∪S(π,Φ). Then by Proposition 49 we have. Let L(s, τ) be the finite part of HeckeL function with respect to τ. Then we have it by Cauchy integral formula.
P(F) -conjugacy Classes
Denote by Cr.e.Pk(F) the union of regular elliptic components of all conjugation classes G(F) in GLk(F),2≤ k ≤ n. LetRkbe a group consisting exactly of all representatives of Pk (F) -conjugation classesCrP.e.k(F). Then the inverse of the elements in the set defined in (4.16) also form a family of representatives of ZG(F)\Cr.e.P(F). Note that these inverses are bijectively P0(F)-conjugate with Re∗.
Holomorphic Continuation
For any given x ∈ G(AF), F1Kχ(x,y) is clearly a well-defined function (with respect to toys) inBloc(ZG(AF)G(F)\G(AF)). Then by Lemma 1.4 in [Art80] (or a more explicit version given by Proposition 13.2 in [Art05], p.71) one sees that given a Siegel set S, positive integers M and M1 and an open compact subgroup K0ofG(AF ,fin), one can choose a finite set{Xi} of left invariant differential operators on ZG(AF,∞)\G(AF,∞) and a positive integer with the property that if (Ω,dω)is a target space andφ (ω) 7→ φ(ω,x)is any measurable function fromΩ toCr(ZG(AF)G(F)\G(AF)/K0), then. Consider naturalSn−2[2]as the stabilizer of {1,2}ofSn. Then Sn−2[2] is clearly isomorphic to Sn−2. Denote by % the natural isomorphism Sn−2. So according to Proposition 49 and Proposition 54 we see that when Re(s) > 1,Ψv(s,Wα,v,Wβ,v;λ)Λv(s, πλ,v ⊗τv× . eπ−λ,v ) −1 are independent of s for all but finitely many placesv, and as a function of s, is a finite product of holomorphic function in Re(s) > 0. Hence both sides of (6.23) are well defined and are meromorphic in Re (s) . So by continuation we have, for Re(s)> 0, that.
LetΦv,l be a constant multiplication of the characteristic function of some open ball inFvn. Then its Fourier transformΦv,l is also of the same form, that is, a constant multiplication of the characteristic function of some open ball inFvn. Let qv = NF/Q(p). Then one has that (see [JPS83]) each(s+λi−λj, σv,i×eσv,j, λ) is of the form cqv−fvs,where|c|= qv1/2and fvis the local conductor, which is bounded by an absolute constant that depends only on Kv-type of the test function ϕ. There exists therefore an absolute constantv ∈N≥0, which only depends on ϕ, so that. F (κ;s). Then one can do the similar analysis to replace κi2,j2 =s−1 withκj2 =s−1. Then we get by induction (or simply continue this process untilm=r−1) the expression (8.14) ).
Letn=4. Then there are three possibilities for r:r =2,r =3 orr =4. We will deal with these cases separately. Then by Corollary 69 we conclude that IWhi(s, τ) has a meromorphic continuation to Re(s) > 0. Then, by functional comparison of Eisenstein series, we conclude that IWhi(s, τ ) has a meromorphic continuation to the whole plane.
Convergence of the Spectral Side
Reduce to the Kuznetsov Relative Trace Formula
Then (5.3) follows from the spectral decomposition K0(x,y) = K(x,y) −K∞(x,y) and the automorphism of these functions with respect to the second variable. So by Cauchy inequality and convolution decomposition of ϕ we get the absolute convergence of IWhi(s, τ). Therefore, IKl(s, τ) is a (multiple) Mellin transform of a Kuznetsov relative trace formula JKuz(ϕ,x), since f(x,s) is essentially |detx|s.
Then IKl(s, τ) is in principle a sum of Kloosterman-sum-zeta functions, which should converge when Re(s) is large enough. But then there exists a constantCv that depends only on N0,v andθ such that θ(xvuvxv−1) = 1 if and only if |αi(xv)|v ≤ Cv,whereαi are the simple roots of G(F) relatively to B is Also, by suppJKuz(ϕ,x;wa) ⊆ A(AF,∞)Aϕ,fin and the compactness of suppϕv, we have kw−1xvwxvakv ≤ Cv0 for some constant Cv0 that depends only on ϕv,v < ∞,andCv0 =1 for almost all.
A straightforward calculation shows that kn1,vkv + kn2,vkv + kyvkv ≤ Cv0 for some constant Cv0 depends only on ϕ. As a consequence of Proposition 36 and the Iwasawa decomposition, we have IKl(s, τ) converges absolutely when Re(s) is large enough.
Spectral Decomsition of the Kernel Function
To demonstrate the absolute convergence of IWhi(s, τ) and thus obtain meromorphic continuation, we need to analyze properties of K∞ by its spectral expansion.
One can write the test function ϕ as a finite linear combination of convolutions ϕ1∗ ϕ2 with functions ϕi ∈ Ccr(G(AF)), whose Archimedean components are differentiable of arbitrarily high order. In fact, the integral overλand sum over χ,Pandφ can be expressed as an increasing limit of nonnegative functions, each of which is at the heart of the restriction of R(ϕj ∗ϕ∗j), a positive semidefinite operator, to an invariant subspace. Note that BP,χ is finite due to the K-finite assumption, and that Eisenstein's series is holomorphic on λ, and therefore becomes the integrand.
For any test function ϕ ∈ H (G(AF), ω), one can write ϕ as a finite linear combination of convolutions ϕj,1 ∗ ϕj,2 with the functions ϕj,i ∈ Ccr(G(AF)) whose Archimedean components are differentiable of arbitrarily high order n,1 ≤ i ≤ 2, and j ∈ J is a finite set. Also in the function field case, the cuspidal datums have no infinitesimal sign, so the sum over χ's is only finite. Note that for any χ and P the space BP,χ depends only on the support and K-finite type of the test function ϕ. Therefore, given any λ◦P =(λ◦.
Note that [IY15] constructed a regularized global integral to calculate these Rankin-Selberg periods in the case of GL(n+1) ×GL(n). Nevertheless, we investigate the analytical behavior of local factors of (5.12) in Section 6, proving that (5.12) is a holomorphic multiple of an L-functionΛ(s, πλ ⊗ τ×eπ−λ).
Discussion on Arthur’s Truncation Operator
Note that (5.14) is absolutely convergent and ΛT is a finite sum for given x andy. One can therefore apply the operators F and ΛT within the integral overia∗P/iaG∗ to obtain this explicitly. Since φv,λ is left N(Fv)-invariant, the right-hand side of the above equality is equal to. Letv ∈ΣF,finbe a finite place such thatπvis is unramified andΦv = Φ◦v is the characteristic function of G(OR,v). Assume thatφ1,v =φ2,v =φ◦v is the unique G(OR,v)-fixed vector in the space ofπvso thatφ0v(e)=1. Then Rv(s,W1,v,W2,v;λ) equals on.
Letv ∈ΣF,∞ be an Archimedean locus such that πv is unramified and Φv = Φ◦v is the characteristic function of G(OR,v). Then one has C∞(σi ⊗ τ× σj;t) ϕ C(σi⊗ τ×σj;t), where the imply constant depends only on suppϕ. So there exists a finite index set. 7.28). Then we can writeRϕ(s,λ;φ)= Rϕ(s,κ;φ)andΛ(s, πλ⊗τ×eπ−λ)=Λ(s, πκ⊗τ×eπ−κ). Remember that if v ∈ΣF, ends a finite place such thatπvis unramified andΦv = Φ◦with the characteristic function of G(OR,v).
F (κ,s) is equal to the multiplication of some holomorphic function. 10.5) This is how we see from ongoing calculations of the analytical behavior of functions. It is clearly seen that the terms on the right-hand side of the above expression are meromorphic in R(1/2), except for the term.
Rankin-Selberg Convolutions for Generic Representations
Let wj be the simple root of GLn corresponding to the permutation (j,j + 1), 1 ≤ j ≤ n−1. Let τn be the longest element in the Weyl group Wno of GLn,n ≥ 2. Since the sum on the right is finite, Rl s,W1,v,W2,v;λ,Φv. is analytic in λ. Moreover, there is a formal Laurent series in {qλvi,qv−λi : 1 ≤ i ≤ r}. That's why part. a) Proposition 49 follows from Proposition 50 below. The functional can be interpreted as the identity between formal Laurent series in {qvλi,qv−λi : 1 ≤ i ≤ r}.
Let α ∈Gm(Fv)n−1 and letWv(α, λ) be a Whittaker function associated withπv,λandα. ThenWv(α, λ) is of the formBv(α, λ)Lv(λ), whereBv(α, λ) )is a holomorphic function, and. It follows from Lemma 48 and induction that Lemma 51 holds for any if it holds forn= 2 case. The general case follows from this and induction, since integral with respect to χl,j is exactly the.
It follows from Lemma 5.4 in [Jac09] that if bothπ are tempered, then the Rankin-Selberg convolution is Ψv s,W1,v,W2,v;λ,Φv. Now we show that Lemma 56 holds forn=2. 6.17) Since the proof is the same, we consider only real places.
From the above formulas and combining with the analytical behavior of the function Resκ1=s−1Resκ2=s−1F (κ,s) we conclude that. One clearly sees that the terms on the right-hand side of the above expression are meromorphic to R(1/2), except for the terms .
Holomorphic Continuation via Multidimensional Residues
Continuation via a Zero-free Region
Therefore, Rϕ(s,κ;φ) as a function of s, is a finite product of holomorphic function in Re(s)> 0; for any given s, such that Re(s) > 0, as a complex function of several variables with respect to κ, Rϕ(s,κ;φ) has the property that Rϕ(s,κ;φ)LS(κ , π ,eπ) is holomorphic, where LS(κ, π,eπ) is denoted by the meromorphic function. Therefore Rϕ(s,κ;φ) is holomorphic in some domain D if LS(κ, π,eπ) does not vanish in D. Now we capture such a zero-free regionD explicitly. So when l > 0, ∂Dχ(l) has two connected components, one of which is exactly the imaginary axis.
Then the proof is similar to that of Theorem H, except that Lemma 37 must be replaced by Proposition 71 and the constant in (7.31) is replaced by meev+1. For each P∈ P, letcP = kP!(2π)kP. Applying Cauchy's integral formula we see that K∞(x,y) is equal to. Substitute (8.13) into (8.11) to obtain an at least formal expansion of bK∞(x,y), which is clearly dominated by the following formal expression.
Note that BP,χ is finite due to the K-finiteness assumption and that the Eisenstein series converge absolutely for our λ, i.e. the integrand. Indeed, the double integral over λ and φ can be expressed as an increasing limit of nonnegative functions, each of which is the kernel of the restriction R(ϕj ∗ϕ∗j), a positive semidefinite operator, to an invariant subspace.
Meromorphic Continuation Inside the Critical Strip
On the one hand, we actually have to rely on Dedekind's conjecture of degreen to be able to process the contribution of the geometric side. On the other hand, when n ≥ 5, the procedure of meromorphic continuation is even more complicated, because we do not have a symmetric description of this process. To prove Theorem I in these cases, we treat n = 3 and n = 4 separately, because we want to provide explicit descriptions.
Proof of Theorem I when n = 3
Then we get a meromorphic continuation inside R(1)− with possible pole ats=1. We will consider these integrals.
Proof of Theorem I when n = 4
Proof of Theorems in Applications
