TRACE FORMULAE FOR SL(2; R) DORIAN GOLDFELD Contents 1. Trace Formula for finite groups 3 2. Trace Formula for the infinite additive group R and subgroup Z 5 3. The Selberg Trace Formula for SL(2; R) (Spectral Side) 7 4. The Selberg Trace Formula for SL(2; R) (Geometric Side) 24 4.1. The Identity Contribution C(Id) 24 4.2. The Contribution C(1) 25 4.3. Contribution of the Hyperbolic Conjugacy Classes C(P ) 31 4.4. Contribution of the Elliptic Conjugacy Class C(R) 35 5. The Selberg Trace Formula for SL(2; R) 39 6. The Kuznetsov Trace Formula for SL(2; R) 42 7. The SL(2; R) Kuznetsov Trace Formula (Spectral Side) 43 8. The SL(2; R) Kuznetsov Trace Formula (Geometric Side) 45 9. The Kuznetsov Trace Formula for SL(2; R) 48 9.1. Kontorovich-Lebedev Transform 48 9.2. Explicit choice of the pT;R test function 49 9.3. Bounds for the pT;R test function 50 9.4. Bounding the Geometric Side of the KTF 54 9.5. Computation of the Main Term M in the KTF 56 9.6. Explicit KTF for GL(2; R) with Error Term 58 9.7. Bounding the Eisenstein term in the KTF 58 10. The Kuznetsov Trace Formula for GL(3; R) 60 10.1. Whittaker functions and Maass forms on h3 63 11. The SL(3; R) Kuznetsov Trace Formula (Spectral Side) 64 11.1. Eisenstein contribution 66 References 69 1 2 DORIAN GOLDFELD 1. Trace Formula for finite groups Let G be a finite group and let Γ be a subgroup of G. We consider the C-vector space of functions n o V := φ : G ! C φ(γx) = φ(x); 8γ 2 Γ; 8x 2 G : We can think of V as the C-vector space of automorphic forms on ΓnG. Definition 1.1. (Inner product on V ) For φ1; φ2 2 V we define X φ1; φ2 := φ1(x)φ2(x): x2ΓnG Definition 1.2. (Orthonormal basis for V ) Let Γz 2 ΓnG and define 1Γz 2 V by ( 1 if x 2 Γz; 1 (x) := Γz 0 otherwise. The functions 1Γz for distinct cosets Γz form an orthonormal basis for V with respect to the above inner product. Definition 1.3. (The kernel function Kf (x; y)) Let G be a finite group and f : G ! C. Let Γ be a subgroup of G. For x; y 2 G we define the kernel function X −1 Kf (x; y) := f x γy : γ2Γ Definition 1.4. (The linear map Kf ) Let φ 2 V . We define a linear map Kf : V ! V where for φ 2 V we have Kf (φ) 2 V where X (Kf φ)(x) := Kf (x; y) · φ(y): y2ΓnG Definition 1.5. (Trace of a linear map on a finite dimensional vector space) Let V be a complex vector space with basis v1; v2; : : : ; vn. A linear map L : V ! V satisfies n n X X L(vi) = ai;jvj; for all 1 ≤ i ≤ n and ai;j 2 C : i=1 j=1 Then associated to L we may define the matrix AL := ai;j 1≤i;j≤n. The trace of L (denoted Tr(L)) is defined to be n X Tr(L) = ai;i; i=1 i.e., it is the trace of the matrix AL. TRACE FORMULAE FOR SL(2; R) 3 Remarks on eigenvalues of a matrix: It is well known that for a matrix A with complex coefficients ai;j; (1 ≤ i; j ≤ n) that • Tr(A) = sum of the eigenvalues of A; • Det(A) = product of the eigenvalues of A. We now compute the action of the linear map Kf : V ! V on the orthonormal basis given in definition 1.2. Let z 2 ΓnG. We have X Kf 1Γz (x) = Kf (x; y) · 1Γz(y) y2ΓnG = Kf (x; z) X = Kf (x1; z) · 1Γx1 (x): x12ΓnG It follows that the matrix AK associated to the linear map Kf is f given by Kf (x1; z) ; and x1;z2ΓnG X Tr(Kf ) = Kf (z; z): z2ΓnG Proposition 1.6. (Trace formula for a finite group G) Let G be a finite group and let Γ be a subgroup of G. Let Cl[Γ] denote the set −1 of conjugacy classes σ γσ σ 2 Γ with γ 2 Γ: For γ 2 Γ we also define −1 Γγ := σ 2 Γ σ γσ = γ ; −1 Gγ := g 2 G g γg = γ : It follows that X #(Gγ) X −1 Tr(Kf ) = f z γz : # (Γγ) γ2Cl[Γ] z2Gγ nG 4 DORIAN GOLDFELD Proof. We have X Tr(Kf ) = Kf (z; z) z2ΓnG X X = f z−1γz z2ΓnG γ2Γ X X X = f z−1 · σ−1γσ · z z2ΓnG γ2Cl[Γ] σ2Γγ nΓ X X X = f z−1 · σ−1γσ · z γ2Cl[Γ] z2ΓnG σ2Γγ nΓ X X = f z−1γz γ2Cl[Γ] z2Γγ nG X X X = f (gz)−1 · γ · gz γ2Cl[Γ] z2Gγ nG g2Γγ nGγ X X X = f z−1 · g−1γg · z γ2Cl[Γ] z2Gγ nG g2Γγ nGγ X X X = f z−1γz γ2Cl[Γ] z2Gγ nG g2Γγ nGγ X #(Gγ) X = f z−1γz : # (Γγ) γ2Cl[Γ] z2Gγ nG The trace formula has two sides: X #(Gγ) X −1 Tr(Kf ) = f z γz : | {z } # (Γγ) spectral side γ2Cl[Γ] z2Gγ nG | {z } geometric side where the geometric side consists of a sum over conjugacy classes. 2. Trace Formula for the infinite additive group R and subgroup Z Consider the additive group G = R and subgroup of rational integers Γ = Z. Following the recipe in x1, we define the C vector space of smooth functions n o V := φ : R ! C φ(x + n) = φ(x); 8n 2 Z; 8x 2 R : TRACE FORMULAE FOR SL(2; R) 5 Definition 2.1. (Inner product on V ) Let φ1; φ2 2 V: We define the inner product Z Z 1 φ2; φ2 := φ1(x) φ2(x) dx = φ1(x) φ2(x) dx: 0 ZnR 2πinx Definition 2.2. (Orthonormal basis for V ) Let en(x) := e : It is well known that every periodic function is a linear combination of en(x) with n 2 Z: Furthermore Z 1 ( 2πi(m−n)x 1 if m = n; em; en = e dx = 0 0 otherwise: This establishes that the en is an othonormal basis of V . n2Z Definition 2.3. (The kernel function Kf (x; y)) Let f : R ! C be a Schwartz function, i.e., a smooth function all of whose derivatives have rapid decay. For x; y 2 R we define the kernel function X Kf (x; y) := f (−x + n + y) : n2Z Definition 2.4. (The linear map Kf ) Let φ 2 V . We define the linear map Kf : V ! V by Z 1 Kf φ (x) := Kf (x; y) · φ(y) dy: 0 Proposition 2.5. (The en are eigenfunctions of Kf ) Let n 2 Z. Then Kf en (x) = fb(−n) · en(x); R 1 −2πixξ where fb is Fourier transform of f given by fb(ξ) := −∞ f(x)·e dx: Proof. Z 1 Kf en (x) = Kf (x; y) · en(y) dy 0 1 Z X = f (−x + n + y) · en(y) dy 0 n2Z 1 1 Z Z = f(−x + y) · en(y) dy = f(y) · en(x + y) dy −∞ −∞ = fb(−n) · en(x); since en(x + y) = en(x) · en(y): 6 DORIAN GOLDFELD Proposition 2.6. (The trace formula for the additive group R) The trace formula for the group R and subgroup Γ = Z is given by X X fb(n) = f(n) : n2Z n2Z | {z } | {z } spectral side geometric side Remark: The trace formula is the well known Poisson summation formula. Proof. We have shown that en is an eigenfunction of the linear map Kf with the eigenvalue fb(n): The trace of Kf is given by the sum of the eigenvalues which is just P fb(n). To find the geometric side we must n2Z compute 1 Z 1 Z X X K(x; x) dx = f(n) dx = f(n): 0 0 n2Z n2Z Note that since the additive group Z is abelian there is only one con- jugacy class, namely Z itself. 3. The Selberg Trace Formula for SL(2; R) (Spectral Side) Let G = SL(2; R) and Γ = SL(2; Z): We also require the maximal compact subgroup cos(θ) sin(θ) K = SO(2; R) = 0 ≤ θ < 2π : − sin(θ) cos(θ) In this case we define the vector space V as a space of smooth functions φ given by n o V := φ : G ! C φ(γgk) = φ(g); 8γ 2 Γ; 8k 2 K; 8g 2 G : By the Iwasawa decomposition it is known that every g 2 G can be uniquely expressed in the form 1 − 1 y 2 xy 2 cos(θ) sin(θ) g = − 1 · 0 y 2 − sin(θ) cos(θ) for some 0 ≤ θ < 2π; x 2 R; y > 0: The action of SL(2; R) on the upper half plane h := fx + iy j x 2 R; y > 0g given by a b az + b z := ; 8 ( a b ) 2 SL(2; ); 8z 2 h ; c d cz + d c d R TRACE FORMULAE FOR SL(2; R) 7 establishes a one-to-one correspondence between G=K and h given by 1 − 1 1 − 1 y 2 xy 2 cos(θ) sin(θ) y 2 xy 2 − 1 · i = − 1 i = x + iy: 0 y 2 − sin(θ) cos(θ) 0 y 2 1 − 1 y 2 xy 2 Notation involving z: We shall use the notation z = − 1 2 0 y 2 G=K and also z = x + iy 2 h interchangeably.