AUTOMORPHIC FORMS ON GL 2

AUTOMORPHIC FORMS ON GL 2 This is an introductory course to modular forms, automorphic forms and automorphic representations. (1) Modular forms (2) Representations of GL 2 (R) (3) Automorphic forms on GL 2 (R) (4) Adèles and idèles (5) Representations of GL 2 (Q p ) (6) Automorphic representations of GL 2 (A) This is a set of notes for my class Automorphic forms on GL(2) in the University of Chicago, Spring 2011. There is obviously no originality in the content and presentation of this very classical materials. 1. Modular forms As usual in representation theory, the letter G is overused. In each chapter, G will denote a different group. In this chapter G = SL 2 (R), K = SO 2 (R), H = G/H is the upper halfplane, D is the open unit disc. Γ will denote a discrete subgroup of SL 2 (R), Γ its image in PGL 2 (R). In particular, Γ(1) = SL 2 (Z) and Γ(1) is its image in PGL 2 (R). 1.1. Geometry of the upper half-plane. The points of projective line are one-dimensional subspaces of a given two-dimensional vector space. The group GL 2 of linear transformations of that two-dimensional vector space thus acts on the corresponding projective line. The action of a 2 2-matrix is given the formula of homographic transformation (1.1.1) a b c d z = az + b cz + d if z denotes the standard coordinate of P 1. This formula is valid for any coefficients fields. In particular, GL 2 (R) acts on P 1 (R) and GL 2 (C) acts compatibly on P 1 (C). It follows that GL 2 (R) acts on the complement of the real projective line inside the complex projective line P 1 (C) P 1 (R) = H H where H (resp. H ) is the half-plane of complex number with positive (resp. negative) imaginary part. Let GL + 2 (R) denote the subgroup of GL 2 (R) of matrices with positive determinant; it is also the neutral component of GL 2 (R) with respect to the real topology. Since GL + 2 (R) is connected, its action on P 1 (C) P 1 (R) preserves H and H. Of course, the above assertion is a consequence of the formula ( ) az + b ad bc (1.1.2) I = cz + d cz + d I(z). 2 1

which derives from a rather straightforward calculation az + b cz + d = (az + b)(c z + d) cz + d 2 = bd + acz z + bc(z + z) + (ad bc)z cz + d 2. This equation becomes even simpler when we restrict to the subgroup SL 2 (R) of real coefficients matrix with determinant one ( ) az + b (1.1.3) I = I(z) cz + d cz + d. 2 From now on in this chapter, we will set G = SL 2 (R). Lemma 1.1.1. The group G acts simply transitively on the upper half-plane H. The isotropy group of the point i H is the subgroup K = SO 2 (R) of rotations : cos θ sin θ (1.1.4) k θ = sin θ cos θ Proof. The equation ai + b ci + d = i implies that a = d, b = c in which case the determinant condition ad bc = 1 becomes a 2 + b 2 = 1. Thus the matrix is of the form (1.1.4). Let z = x + iy with x R and y R +. It is enough to prove that there exists a, b, c, d R with ad bc = 1 such that ai + b ci + d = z. We set c = 0. We check immediately that the system of equations ad = 1, a = yd, b = xd has real solutions with d = y 1/2, a = y 1/2 and b = xy 1/2. We observe that this calculation shows in fact G = BK where B is the subgroup of G consisting of upper triangular matrices. This is a particular instance of the Iwasawa decomposition. Lemma 1.1.2. The metric (1.1.5) ds 2 = dx2 + dy 2 on H, as well as the density µ = dxdy/y 2 is invariant under the action of G. a b Proof. With the notations γ = and z c d = γz, we have (1.1.6) dz = y 2 (ad bc) (cz + d) 2 dz. This calculation has the following concrete meaning. The smooth application g : H H maps z z. It induces linear application on tangent spaces T z H T z H and its dual linear application T z H T zh. The cotangent space T z H (resp. T zh) is a onedimensional C-vector space generated by dz (resp. dz). The linear application sends dz on ((ad bc)/(cz + d) 2 )dz. 2

The element dz induces the canonical quadratic form dx 2 + dy 2 on T z H viewed as 2- dimensional real vector space. Similarly, we have the quadratic form dx 2 + dy 2 on T z H. The equation (1.1.6) implies that dx 2 + dy 2 = (cz + d)2 cz + d 4 (dx2 + dy 2 ). It follows that the metric ds 2 = (dx 2 + dy 2 )/y 2 is invariant under G, according to (1.1.2). The same argument applies to the density µ = dxdy/y 2. Lemma 1.1.3. The Cayley transform (1.1.7) z cz = 1 i z = z i 1 i z + i. maps isomorphically H onto the unit disk D = {z C z < 1}. The inverse transformation is (1.1.8) w c 1 w = 1 1 1 i(1 + w) w = 2 i i 1 w. The metric ds 2 = (dx 2 + dy 2 )/y 2 on H transports on the metric (1.1.9) d D s 2 = 4(du2 + dv 2 ) (1 w 2 ) 2 where w = u + iv. We also have (1.1.10) dxdy/y 2 = 4dudv (1 w 2 ) 2. Proof. See [5, Lemma 1.1.2] Since c and c 1 are inverse functions of each other, it is enough to check that c(h) D and c 1 (D) H. For every z H, we have z i < z + i so that c(z) < 1. It follows that c(h) D. For every w D, the straightforward calculation (1.1.11) shows i(1 + w) 1 w = 2I(w) + i(1 w 2 ) 1 w 2 (1.1.12) y = 1 w 2 1 w 2 > 0. if z = c 1 w and y = I(z). It follows that c 1 (D) H. By using the chain rule we have dz = 2idw (1 w). 2 If we write w = u + iv in cartesian coordinates, then we have dx 2 + dy 2 = 4(du2 + dv 2 ). 1 w 4 It follows that dx 2 + dy 2 = 4(du2 + dv 2 ) y 2 (1 w 2 ). 2 The same calculation proves the expression of the measure on the disc (1.1.10). 3

Lemma 1.1.4. Any two points of H are joined by a unique geodesic which is a part of a circle orthogonal to the real axis or a line orthogonal to the real axis. Proof. See [5, Lemma 1.4.1]. Instead of H we consider the unit disc. We assume that the first point is 0 and the second point is a positive real number a < 1. Let φ : [0, 1] D with φ(t) = (x(t), y(t)) denote a parametrized joining 0 = (x(0), y(0)) and a = (x(1), y(1)). Its length is that is at least 1 0 1 2(1 φ(t) 2 ) 1 (dx(t)/dt) 2 + (dy(t)/dt) 2 dt 0 2(1 x(t) 2 ) 1 dx(t)/dt dt = a 0 2dt (1 t 2 ) The shortest curve joining 0 and a is thus a part of a radius in the unit disc. For every two points x 0, x 1, there is g SL 2 (R) that maps H on D, cg(x 0 ) = 0 and cg(x 1 ) = a where a is a positive real number satisfying a < 1. Here c : H D is the Cayley transform. The geodesic joining x 0 with x 1 is a part of the preimage of the radius from 0 to a. That preimage is necessarily part of a circle or a strait line. Moreover as the transformation cg is conformal, that circle or line must be orthogonal with the real line as the radius [0, a] is orthogonal to the unit circle. Exercice 1.1.5. [2, Ex. 1.2.5] Let SL(2, C) acts of bp 1 (C) by the homographic transformation (1.1.1). Prove that the subgroup that map the unit disc D onto itself is { } a b (1.1.13) SU(1, 1) = a b 2 b 2 = 1. ā Prove that the subgroup SU(1, 1) is conjugate to SL(2, R) in SL(2, C). Prove that the subgroup of SU(1, 1) that fixes 0 D is the rotation group {} e iθ 0 0 e iθ. 1.2. Fuschian groups. We will be mainly interested on the quotient of H by a discrete subgroup of G. The most important examples of discrete subgroups are the modular group SL 2 (Z) and its subgroup of finite indices. We will call Fuchsian group a discrete subgroup of SL 2 (R). Proposition 1.2.1. A Fuchsian group Γ acts properly on the upper half-plane H. Proof. Recall that the action Γ on H is proper means that the map Γ H H H defined by (γ, x) (x, γx) is proper i.e. the preimage of a compact is compact. We need to prove that for every compact subsets U, V H, the set {γ Γ γu V } is a finite. Because the group G = SL 2 (R) acts on H with compact stabilizer, the subset {g G γu V } is compact. Its intersection with the discrete subgroup Γ is finite. Corollary 1.2.2. For every Fuchsian group Γ, the quotient Γ\H is a Haussdorf topological space. 4

An element g SL 2 (R) is called elliptic if it has a fixed point in H. It follows from the above that g is elliptic if and only if it is conjugate to an element of SO(2, R) cos θ sin θ sin θ Any element g SL 2 (C) has at least on fixed point in P 1 (C). If g SL 2 (R) is not elliptic, it must have fixed points on P 1 (R) = R { }. We call g parabolic if it has a unique fixed point, and hyperbolic if it has two fixed points. A parabolic element is conjugate to a matrix of the form ɛ x (1.2.1) 0 ɛ cos θ with ɛ {±1}. A hyperbolic element is conjugate to a diagonal matrix a 0 (1.2.2) 0 a 1 with a R. Let Γ be a Fuschian group. For every x H, the stabilizer Γ x of x in Γ is a finite group because it is the intersection of a compact group with a discrete group. In fact, since SO 2 (R) is isomorphic to the circle, for every x H, Γ x is a finite cyclic group. If Γ x is nontrivial, we call x an elliptic point of Γ. This also implies taht elliptic points are isolated. Lemma 1.2.3. There is a canonical complex structure on Γ\H so that the quotient map H Γ\H is complex analytic. Proof. If [z] is the Γ-orbit of z H such that Γ z = Γ Z. Then there exists a neighborhood U of z consisting of points with the same property. This neighborhood is homeomorphic with its image in Γ\H. Its image in Γ\H is thus equipped with an analytic structure inherited from U. If z is the Γ-orbit of z H such that Γ x = Γ Z such that Γ z is a finite group larger than Γ Z. Γ z is then a finite cyclic group µ k. By a homographic transformation we can change the model from H to D and maps z to 0. The quotient of a small disc around 0 by the action of µ k can be given a complex structure with uniformizing parameter w d where w is the standard uniformizing parameter of D around 0. This defines a complex structure on a neighborhood of [z] in Γ\H. Definition 1.2.4. A point x P 1 (R) is called a cusp for Γ if it is the fixed point of a nontrivial parabolic element. In that case Γ x is isomorphic to the product of Z with an infinite cyclic group, the first factor Z = µ 2 being the center of G. Let P Γ denote the set of cusps of Γ. We set H = H P Γ. We consider the topology on H by adding to the real topology on H a family of neighborhoods of each cusp x P Γ. If x =, we take the family (1.2.3) U l = U l { } with U l = {z H I(z) > l} The family of neighborhoods of other cusps are constructed from the U l by conjugation. Lemma 1.2.5. Γ\H is a Haussdorff space. 5

Proof. See [5, Lemma 1.7.7]. As we already know that Γ\H is Haussdorff, it remains to prove that a cusp and a point of H are separated and two cusps are separated that can be checked directly upon the definition. Lemma 1.2.6. There is a complex analytic structure on Γ\H that extends the complex analytic structure on Γ\H. Proof. We can restrict ourselves to the case that is a cusp and to define an analytic structure around the image of in Γ\H. The stabilizer of in G is a b (1.2.4) G = { 0 a 1 a R, b R} By definition, is a cusp of Γ if Z(Γ G ) is a subgroup of the form ɛ mn (1.2.5) { ɛ {±1}, m Z} 0 ɛ for some fixed integer n. The map z e 2iπz/n defines a homeomorphism from (G Γ)\Ul where Ul is a standard neighborhood (1.2.3) of H on a disc centered at 0. This provides (G Γ)\H with a complex analytic structure. Definition 1.2.7. A discrete subgroup Γ of SL 2 (R) is called a Fuchsian group of first kind if X Γ = Γ\H is compact. Proposition 1.2.8. If X Γ is compact, then the numbers of elliptic points and cusps of Γ in Γ\H are finite. Theorem 1.2.9 (Siegel). A discrete subgroup of SL 2 (R) is a Fuchsian group of first kind if and only if Γ\H has finite area. Proof. We refer to [5, Theorem 1.9.1] for the proof of this theorem. We will only be interested in the case of arithmetic groups in which the conclusion of the theorem can be established directly by other means. A connected domain F of H is called a fundamental domain of Γ if F satisfies the following conditions (i) H = γ Γ γf ; (ii) if U is the set of interior points of F then F = Ū ; (iii) γu U = for all γ Γ not belonging to the center Z of G. Lemma 1.2.10. Every Fuchsian group has a fundamental domain. Proof. An element γ Γ Z only has finitely many fixed points. Since Γ is countable, there exists z 0 H which is not fixed by any element γ Γ Z. For every γ Γ, we put F γ = {z H d(z, z 0 ) d(z, γz 0 )} U γ = {z H d(z, z 0 ) < d(z, γz 0 )} C γ = {z H d(z, z 0 ) = d(z, γz 0 )} Here, d indicates the hyperbolic distance on H defined by the metric ds 2 = (dx 2 + dy 2 )/y 2. The intersection F = γ Γ Z 6 F γ

is a fundamental domain of Γ. We will now review the classification of 2 2 real matrices up to conjugation. A matrix is said to be : (1) hyperbolic if its has distinct real eigenvalues; (2) elliptic if it has distinct complex conjugate eigenvalues; (3) parabolic if it is not central and and has an eigenvalue of multiplicity two; (4) central otherwise. Lemma 1.2.11. Let γ SL 2 (Q) act on P 1 (C) by homographic transformation. Let z C so that γz = z. If γ is hyperbolic (resp. elliptic) then z is either rational or generates a real (resp. imaginary) quadratic extension of Q. If γ is parabolic then z Q. Among the Fuschian groups, we are particularly interested in the modular group SL 2 (Z) and its subgroups of congruence (1.2.6) (1.2.7) (1.2.8) Γ(N) = {γ SL(2, Z) γ 1 mod N} 1 Γ 1 (N) = {γ SL(2, Z) γ mod N} 0 1 Γ 0 (N) = {γ SL(2, Z) γ mod N} 0 In particular, we will use the convenient notation Γ(1) = SL 2 (Z) for the full modular group. We consider the case of the full modular group. Let F denote the domain defined by the conditions R(z) 1/2 and z 1. Consider the two matrices of Γ(1) 1 1 0 1 (1.2.9) T = and S = 0 1 1 0 They act on H by the following rules T z = z + 1 and Sz = 1/z. Lemma 1.2.12. Let Γ denote the subgroup of Γ(1) generated by the transformations S and T as above. (1) For every z H, there exists γ Γ such that γz F. (2) If z, z F and γ Γ(1) non trivial such that γz = z then z, z both lies in the boundary of F. (3) F is a fundamental domain for Γ(1), and Γ(1) is generated by the matrices S and T. Proof. For every z H, the lattices generated 1 and z have only finitely many members cz + d such that cz + d 1. It implies that there are only finitely many z = γz conjugate to z such that I(z ) I(z). We can assume that I(z) is maximal among all the conjugates γz with γ Γ. With the help of the translation T, we can assume that z belongs the the vertical strip R(z) 1/2. We only need to prove that under these assumptions, we have z 1. If z < 1, we would have I( 1/z) > I(z) that would contradict the maximality of I(z). It follows that z F. Let z, z F and a b γ = Γ(1) such that z c d = γz. 7

We can assume that I(z) I(z ). This implies that cz + d 1. By a careful inspection, this implies in particular that z must lie on the boundary of F. More inspection shows that z lie also on the boundary of F. See [6, p.130]. Let z U be an element of the interior of F. For every γ Γ, there exists γ Γ such that γ γz F. The assumption that z lie in the interior of F implies γ γ = 1, thus γ Γ. We have also checked all the conditions that makes F a fundamental domain of Γ(1). Proposition 1.2.13. For Γ(1) = SL(2, Z), the set of cusps is Q { }. They are all conjugate under the action of Γ(1). The quotient X Γ(1) = Γ(1)\H is isomorphic to P 1 (C) as complex analytic space. Up to equivalence, Γ(1) has one elliptic point of order 2 that is i H with b 2 = 1 and one elliptic point of order 3 that is j H with j 3 = 1. Proof. An element x R which fixed by a parabolic matrix γ Γ(1) must be a rational. It follows from the shape of the fundamental domain F of Γ(1) that Γ(1)\H is a compact Riemann surface that is homeomorphic to the sphere. It is isomorphic to P 1 (C). Corollary 1.2.14. Congruence subgroups are Fuchsian groups of first kind who set of cusps is Q { }. There are only finitely many cusps up to the action of Γ. Proof. Since Γ is a subgroup of Γ(1) with finite index, they have the same set of cusps Q { }. Since Γ(1) acts transitively on this set, the number of Γ-orbits in this set is at most equal to the index of Γ in Γ(1). Since Γ(1)\H is compact, and Γ is a subgroup of Γ(1) of finite index, the quotient Γ\H is also compact. Let x n be a sequence of points of Γ\H. We need to prove that there exists a convergent subsequence. Let z n be a sequence in H so that x n = Γz n is the image of z n in Γ\H. Let x i denote the image of z n in Γ(1)\H. Since Γ(1)\H is compact, we can assume that x n converges the x Γ(1)\H. Let z H be a preimage of x. There exists γ n Γ(1) so that γ n z n converges to z. Because Γ/Γ(1) is finite, after extracting a subsequence, we can assume that there exist γ Γ(1) so that γ n γγ(1) for all n. It follows that x n converges to γ 1 x where x is the image of z in Γ\H. This proves that Γ is a Fuschian group of first kind. Proposition 1.2.15. Let Γ be a subgroup of Γ(1) of finite index µ. Let m 2, m 3 be the number of Γ-equivalence classes of elliptic points of orders 2 and 3 respectively. Let m denote the number of Γ-equivalence classes of cusps. Then the genus of Γ\H is g = 1 + µ 12 m 2 4 m 3 3 m 2. Proof. See [7, 1.40]. The map Γ\H Γ(1)\H is a finite proper map of degree µ = [ Γ(1) : Γ]. This is an application of Hurwitz formula. As the morphism Γ\H Γ(1)\H is of degree µ and Γ(1)\H is of genus 0, we have 2g 2 = 2µ + (e P 1) P with P in the set of ramified points, e P being the index of ramification. Summing e P over the ramified points P over j we get 2(µ m 3 )/3. The same sum over i is (µ m 2 )/2 and 8

over is µ m. By summing altogether, we get the desired formula for the genus of Γ\H. Corollary 1.2.16. If Γ do not have elliptic points then the genus of Γ\H is g = 1 + µ 12 m 2. Exercice 1.2.17. [2, p.24] (1) Prove that a fundamental domain for Γ(2) consists of x + iy such that 1/2 < x < 3/2, z + 1/2 > 1/2, z 1/2 > 1/2 and z 3/2 > 1/2. (2) Prove that Γ(2) is generated by the matrices 1 2 1 0 and 0 1 2 1 (3) Prove that Γ(2) has three inequivalent cusps and Γ(2)\H is isomoprhic to P 1 (C) (4) Prove that if φ is an entire function such that there exists two distinct complex numbers a, b that don t belong to the image of φ then φ is a constant function (Picard s theorem). 1.3. Modular forms. Let k be an even nonnegative integer. A modular form of weight k for Γ = SL(2, Z) is a holomorphic function on H which satisfies the identity ( ) az + b (1.3.1) f = (cz + d) k f(z) cz + d for all z H and a b SL(2, Z) c d and which is holomorphic at the cusp. The last condition requires some discussion. We have define the analytic structure of Γ\H near by choosing as the local coordinate the function q = e 2πiz. The equation 1.3.1 implies in particular f(z + 1) = f(z), and thus f has a Fourier expansion (1.3.2) f(z) = n Z a n e 2iπnz = n Z a n q n. The function f is holomorphic at the cusp if in the above expansion a n = 0 for n < 0. If furthermore a 0 = 0, we say that f is cuspidal at. If Γ is a Fuschian group of first kind, we can also modular forms of weight k for Γ similarly. The holomorphic function f on H is required to satisfy the same equation (1.3.1) and to be holomorphic at the cusps of Γ. If x is a cusp, the stabilizer of x in PGL 2 (Z) is the infinite cyclic group generated by a parabolic element. After conjugation, we can assume that the cusp is the point and its stabilizer in Γ is generated by the matrix 1 m (1.3.3) 0 1 for some positive real number m R. The holomorphicity at this cusp is equivalent to that f admits a Fourier expansion f = n Z a n q n with a n = 0 for n < 0 with respect to the variable q = e 2iπz/m. 9

Lemma 1.3.1. Suppose that is a cusp of a Fuschian group Γ of first kind with Γ generated by the matrix (1.3.3). Let f be a modular form of weight k and let a n q n with q = e 2iπz/m denote its Taylor expansion near this cusp. Then the series a n q n converges absolutely and uniformly on every compact in H. Proof. The function z q = e 2iπz/m defines an isomorphism between Γ \H and the punctured disc D {0}. By assumption modular form f defines a holomorphic function on D {0} that extends holomorphically to D. This implies that the Taylor series n=0 a nq n converges absolutely uniformly on every compact contained in D. Definition 1.3.2. Let Γ be a Fuschian group of first kind. We denote M k (Γ) the space of modular forms of weight k for Γ. We denote by S k (Γ) the space of cusp forms of weight k for Γ. We also denote A k (Γ) the space of meromorphic functions f of H satisfying (1.3.1) that are meromorphic at the cusps. Proposition 1.3.3. If k = 0, A 0 (Γ) is the field F Γ of meromorphic functions on X Γ. We have M 0 (Γ) = C and S 0 (Γ) = 0. We will now determine the dimension of M k (Γ) and S k (Γ) for even integers k. We will refer to [7, 2.6] and [5] for the case of odd integers. The case k = 1 does not seem to be treated so far. Proposition 1.3.4. There is a canonical isomorphism between the space A 2k (Γ) of meromorphic automorphic forms of weight 2k and the space Ω k X Γ OXΓ F Γ of meromorphic k-fold differential form on X Γ. In particular, A 2k (Γ) is a one-dimensional F Γ -vector space. Proof. For every f A k (Γ), the k-fold differential form f(z)(dz) k is Γ-invariant. It descends to a meromorphic k-fold differential form ω f on Γ\H. The condition of meromorphicity of f at the cusps impies that ω f is a meromorphic form on X Γ. The application f ω f induces an isomorphism A 2k (Γ) Ω k X Γ OXΓ F Γ. In order to calculate the dimension of M k (Γ), we will express the condition of holomorphicity of f in terms of the zero divisor of ω f on X Γ and then apply the theorem of Riemann-Roch. This calculation will be done separately in three cases : general points, elliptic points and cusps : Let z H be a non-elliptic point with image x X Γ. The function f is holomorphic at z 0 if and only if ω f is holomorphic at x. Let z H be an elliptic point of index e and let denote x X Γ its image. Let t be a local parameter at z H and u a local parameter at x X Γ. We have t e u where the equivalence means equal up to an invertible function on a neighborhood of z. By derivation, we have du z e 1 dz. Raising to the power k, we have (dz) k z k(e 1) (du) k. Let denote ν x (f) the valuation of f with respect to the parameter u i.e. ν x (u) = 1 and ν x (t) = 1/e. Let us calculate the order of vanishing of the k -fold differential form ω f = f(dz) k fz k(e 1) (du) k at x. We have ord x (ω f ) = ν x (f) k(1 1/e). The function f is holomorphic at z if and only if ν x (f) 0 which is equivalent to ord x (ω f ) + k(1 1/e) 0. 10

Since ord x (ω f ) is an integer, the above inequality is equivalent to ord x (ω f ) + [k(1 1/e)] 0 where as usual [r] denotes the largest integer that is not greater than a given real number r. In the case of weight two form i.e. k = 1, the above condition means simply that ord x (ω f ) 0. In the general case the integer e cannot be ignored. Let us consider a cusp of Γ that we can assume to be without loss of generality. Let us denote x its image in Γ\X Γ. The development of f at the cusp has the form f(z) = n Z a n q n where q = 2iπmz for some positive integer m. Let r be the least integer such that a r 0. We note ν x (f) = r. Let us denote ω f = fdz k. Since dz dq/q we have ω f fq k dq k. By construction, q is a local parameter of X Γ at x. It follows that ν x (ω f ) = ν x (f) k. Thus f is holomorphic at the cusp i.e. ν x (f) 0 if and only if ν x (ω f ) + k 0 and f vanishes at the cusp i.e. ν x (f) 1 if and only if ν x (ω f ) + k 1. In wight two case k = 1, f is holomorphic at if ω f is a logarithmic one form and f is a cusp form if and only if ω f is a holomorphic one form. Let denote x its image in Γ\H and let choose a local parameter u of x Γ\H. Proposition 1.3.5. Let Γ be a Fuschian group of first kind. The space M 2 (Γ) is canonically isomorphic with the space H 0 (X Γ, Ω XΓ (cusp)) of one form with logarithmic singularities at the cusps. The space S 2 (Γ) is canonically isomorphic with the space of holomorphic one form of X Γ S 2 (Γ) = H 0 (X Γ, Ω XΓ ). In particular dim S 2 (Γ) = g (calculated in 1.2.15) and dim M 2 (Γ) = g + m 1 where g is the genus of X Γ and m is the number of inequivalent cusps. Proposition 1.3.6. Let Γ be a Fuschian group of first kind and let 2k be an even integer greater or equal to 4. We have and dim M 2k (Γ) = dim S 2k (Γ) + m. dim S 2k (Γ) = (2k 1)(g 1) + s [k(1 1/e i )] + (k 1) where x 1,..., x s denote the elliptic points, e 1,..., e s their elliptic index and m is the number of inequivalent cusps. In absence of elliptic points, we have i=1 dim S 2k (Γ) = (2k 1)(g 1) + (k 1)m. In the case Γ = Γ(1), we have g = 0, m = 1 and two elliptic points with indexes {2, 3}. 11

Corollary 1.3.7. We have 0 if k = 1 dim S 2k (Γ(1)) = [k/6] 1 if k > 1 and k 1 mod 6 [k/6] otherwise and { 0 if k = 2 dim M 2k (Γ(1)) = dim S 2k (Γ(1)) + 1 otherwise In particular dim M 4 (Γ(1)) = dim M 6 (Γ(1)) = 1. We can construct an explicit generator for these spaces by Eisenstein series. Let 2k be an even integer with 2k 4. Define (1.3.4) E 2k (z) = 1 (mz + n) 2k 2 (m,n) Z 2 (0,0) This series is absolutely uniformly convergent on compact domain and defines a holomorphic function on H. This function is a modular form of weight 2k for the full modular group Γ = SL(2, Z). The automorphy (1.3.1) dervies from the action of SL(2, Z) on the set Z 2 (0, 0). We will prove in 1.4.1 that Eisenstein are holomorphic at the cusp. In particular, it will be proved that the free coefficient in the Fourier expansion of E 2k is ζ(2k). We will choose a normalization so that the free coefficient be one G 2k (z) = ζ(2k) 1 E 2k. The space M 4 (Γ(1)) is generated by G 4, M 6 (Γ(1)) is generated by G 6, M 8 (Γ(1)) is generated by G 2 4, M 10 (Γ(1)) is generated by G 4 G 6. In weight 12 there is the first cusp form = (G 3 4 G 2 6)/1728. Proposition 1.3.8. The rational function j : G 3 4/ defines an isomorphism from X Γ(1) onto P 1 C Proof. By the discussion that precedes 1.3.5, there is a line bundle L on X Γ (1) such that M 12 (Γ(1)) = H 0 (X Γ(1), L). We know dim H 0 (X Γ(1), L) = 2. Since X Γ(1) = P 1, L = O P 1(1). It follows that G 3 4 and as global section of L vanish exactly at one point. Morever as they are not proportional, their quotient define a morphism that is an isomorphism. j : G 3 4/ : X Γ(1) P 1 C 1.4. Fourier coefficients of modular forms. We have an explicit formula for Fourier coeffients of Eiseinstein series Proposition 1.4.1. The Fourier expansion of E k has the form (1.4.1) E k (z) = ζ(z) + (2πi)k σ k 1 (n)q n. (k 1)! where σ k 1 (n) = d n dk 1. 12 n=1

Proof. See [2, p.28] The terms with m = 0 in LHS sum up to the term ζ(z) in RHS. We have 1 n k = n k = ζ(k). 2 n Z {0} n N In order to deal with the other terms, we will need the following lemma. Lemma 1.4.2. Let k be an integer greater or equal to two. We have the formula (1.4.2) (n z) k = (2πi)k n k 1 e 2πinz (k 1)! for all z H. n Z Proof. See [2, p.12] for more details. For a fixed z, the function f(x) = (x z) k is a complex analytic function with a pole at x = z. On the real line, it has no pole if I(z) > 0 and it is L 1 if k 2. Its Fourier transform is given by the formula ˆf(y) = n N (x z) k e 2iπxy dx. We can evaluate this integral by applying the residue formula to the 1-form (x z) k e 2iπxy dx. We get { 2πi res x=z ((x z) ˆf(y) k e 2πixy dx) if y > 0 = 0 if y 0. The calculation of the residue gives ˆf(y) = { (2πi) k (k 1)! yk 1 e 2πiyz if y > 0 0 if y 0. We apply now the Poisson summation formula reviewed in B.2.2. In (1.3.4), the terms with a fix m > 0 and and thoese with its opposite are equal. By taking the factor 1/2 into account, we only need to consider the terms with m > 0. Apply the above lemma to mz, we will get (1.4.3) n Z (mz + n) k = (2πi)k (k 1)! n k 1 e 2πimnz. If we sum the above formula over the positive integers m, we will get (1.4.1). Corollary 1.4.3. Let E k (z) = n=0 a nq n be the Fourier expansion of the Eisenstein series at. There exists positive constants A, B > 0 such that An k 1 a n Bn k 1 for every n N. Proof. By the formula (1.4.1), it is enough to seek such an estimation for σ k 1 (n). In one side we have the obvious inequality n k 1 σ k 1 (n). On the other hand, we have the inequality σ k 1 (n) n k 1 = d n n N 1 ζ(k 1) dk 1 valid for k > 2. 13

Proposition 1.4.4. If f = n=1 a nq n is a cusp form of weight k, its Fourier coefficients satisfy the inequality a n Cn k/2 for some constant C independent of n. Proof. The equations (1.1.2) and (1.3.1) imply that the continuous function f(z)y k/2 on H is Γ-invariant. It so defines a continuous function on the quotient Γ\H. Assume now Γ = SL 2 (Z). Since q = e 2πy, the vanishing of the constant terms of the Fourier expansion of f in the variable q implies that lim y f(z)y k/2 = 0. It follows that the function f(z)y k/2 is bounded by a constant C 1. For every natural integer n N, and for every y > 0, we have a n e 2πny = 1 Let us pick y = 1/n and derive the inequality with C = e 2π C 1. 0 f(x + iy)e πinx dx C 1 y k/2. a n < Cn k/2 This bound can be improved according to the Ramanujan-Peterson conjecture. Theorem 1.4.5. Let f = n=1 a nq n is a cusp form of weight k of level N. Then for (n, N) = 1, we have a n = O(n k 1 2 ). This conjecture was proved by Eichler, Shimura and Igusa in the case k = 2. The proof in the k > 2 is due to Deligne. It is based on the Eichler-Shimura relation and the Weil conjecture. Among cusp forms, the eigenvectors with respect to the Hecke operators that we will later introduce, have Fourier coefficients with arithmetic significance. In particular, since the space S 12 (Γ(1)) of cusp forms of weight 12 for Γ(1) is one dimensional, its generator is automatically an eigenvector. The function of Ramanujan (z) = n=1 τ(n)qn is a normalized cusp form whose Fourier coefficients are integers. Deligne proved the inequality that is the original conjecture of Ramanujan. τ(p) 2p 11/2 1.5. L-function attached to modular forms. If f M k (Γ) is a modular form with Fourier expansion f = n=1 a nq n. We call the Dirichlet series (1.5.1) L(s, f) = a n n s The bounds on the Fourier coefficients 1.4.3 and 1.4.4 implies that this Dirichlet series converges on a half-plane. We also consider the complete L-function n=1 (1.5.2) Λ(s, f) = (2π) s Γ(s)L(s, f). Hecke s theory takes a rather simple form in the case of the full modular group Γ(1). Proposition 1.5.1. Suppose that f is a modular form of weight k for Γ(1). If f is a cusp form, Λ(s, f) extends to an analytic function of s, bounded on vertical strips. If f is not a cusp form, then Λ(s, f) extends to a meromorphic function with simple poles s = 0 and s = k. It satisfies the functional equation (1.5.3) Λ(s, f) = i k Λ(k s, f). 14

Proof. We will restrict ourselves to the case of a cusp form for the full modular group. Because f is cuspidal, f(iy) 0 vary rapidly as y. We use the automorphy equation (1.3.1) for the element 0 1 S = Γ(1) 1 0 and derive the equality (1.5.4) f(iy) = i k y k f(i/y) It follows that f(iy) 0 very rapidly as y 0 too. It follows that the integral (1.5.5) 0 0 f(iy)y s dy y is convergent for all s and defines an analytic function of s. The following Mellin integral is absolutely convergent for R(s) > ν + 1 f(iy)y s dy (1.5.6) = a n e 2nπy y s dy y y (1.5.7) (1.5.8) (1.5.9) = 0 1 a n (2nπ) s 1 = (2π) s Γ(s) = Λ(s, f) 0 a n n s 1 e y y s dy y The exchange of the integration and infinite series is licit because the series 1 a ne 2nπy is absolutely convergent as well as 1 a nn s. It follows that the expression 1.5.5 defines an analytic continuation of Λ(s, f). The functional equation (1.5.3) derives from the substitution of y by 1/y in (1.5.5). The converse theorem is also easy in the case of Γ(1). Proposition 1.5.2. Let a 1, a 2... be a sequence of complex numbers which is O(n ν ) for some positive real number ν. Let L(s, f) be defined by the series (1.5.1), convergent for R(s) > ν + 1. Let Λ(s, f) be the function defined by (1.5.2) and suppose it has an analytic continuation for the complex plan of s which is bounded in any vertical strips. Assume that Λ(s, f) satisfies the functional equation (1.5.3) for some positive integer k. Then f(z) = n=1 a nq n is a cusp form of weight k for Γ(1). Proof. The assumption a n = O(n ν ) implies that the series n=1 a nq n is absolutely for q < 1 or over the half-plane z = H if q = e 2πiz. The Mellin integral (1.5.10) (1.5.11) (1.5.12) 0 f(iy)y s dy y = = 0 1 a n e 2nπy y s dy y a n (2nπ) s 1 = Λ(s, f) 15 0 e y y s dy y

is absolutely convergent for R(s) > ν+1. Let σ be a positive real number such that σ > ν+1. The inverse Mellin transform will permit us to recover the value of f on the imaginary axis ir from Λ(s, f) (1.5.13) f(iy) = 1 2πi t= Λ(σ + it, f)y σ it dt. The convergence of this integral follows from the absolute convergence of the Dirichlet series and the Stirling formula for the Gamma function: (1.5.14) Γ(s) 2πe s s s 1/2 as s and R(s) δ > 0. On the vertical line s = σ + it for fixed σ > 0, we have Γ(σ + it) 2π t σ 1/2 e π t /2 as t. With the functional equation (1.5.3), we can transform (1.5.13) as follows (1.5.15) (1.5.16) f(iy) = i k 1 2πi = i k y k 1 2πi t= Λ(k σ it, f)y σ it dt t = Λ((k σ) + it, f)y k σ+it dt after the change of variable t = t. We note that the line R(s) = k σ is out of the convergence domain on the Dirichlet series and we would like to move back the line of integration to the domain of convergence of the Dirichlet series. In order to apply Cauchy theorem, we need to prove that Λ(x + iy, f) 0 as y ± uniformly when x varies in a compact set. For a fixed x > σ, this follows again from the absolute convergence of the Dirichlet series and the Stirling formula for the Gamma function.for a fixed x << 0, we obtain the same convergence Λ(x + iy, f) 0 as y ± by using the functional equation (1.5.3). The uniform convergence follows from an application of the Phragmén-Lindelof principle. Proposition 1.5.3. Let f(s) be a holomorphic function on the upper part of a strip defined by the inequalities σ 1 R(s) σ 2 and I(s) > c. Suppose that f(σ + it) = O(e tα ) for some real number α > 0. Suppose that for some M R, f(σ + it) = O(t M ) for σ = σ 1 or σ = σ 2. Then f(σ + it) = O(t M ) uniformly in σ [σ 1, σ 2 ]. Proof. See [5, p.118] and [10, p.124]. We can now apply the Cauchy theorem and move back the integration line to R(s) = σ (1.5.17) f(iy) = i k y k 1 2πi and apply again the inverse Mellin transform t= (1.5.18) f(iy) = i k y k f(iy 1 ). Λ(σ + it, f)y σ+it dt This is the automorphy relation (1.3.1) with respect to the element S of (1.2.9). The automorphy relation for T requires no proof since f is defined as a Fourier series. As S and T generates the full modular group Γ(1), f is a cusp form of weight k for Γ(1). 16

The case of a general congruence group Γ Γ(1) is considerably more complicated, mainly because the matrix S fails to belong to Γ. As we will see later 1.7, it it enough to restrict to congruence subgroup of the form Γ(N) Γ 1 (N) Γ 0 (N) with (1.5.19) (1.5.20) (1.5.21) For every N, there is an exact sequence Γ(N) = {γ SL(2, Z) γ 1 mod N} 1 Γ 1 (N) = {γ SL(2, Z) γ mod N} 0 1 Γ 0 (N) = {γ SL(2, Z) γ mod N} 0 1 Γ 1 (N) Γ 0 (N) (Z/NZ) 1. It follows that the group (Z/NZ) acts on the space of modular forms M k (Γ 1 (N)) as well as the space of cusp forms S k (Γ 1 (N)). For every primitive character χ : (Z/NZ) C, we will denote (1.5.22) M k (N, χ) = {f M k (Γ 1, N) cf = χ(c)f c (Z/NZ) }. Similar notation S k (N, χ) for cusp forms will also prevail. We will call these forms modular forms of level N of nebentypus χ. We will need the following lemma that replaces the role of the element S Γ(1). Lemma 1.5.4. The matrix (1.5.23) S N = 0 1 N 0 normalizes Γ 0 (N). Furthermore, it transforms the space S k (N, χ) into S k (N, χ) where χ is the opposite Dirichlet character of χ. Theorem 1.5.5 (Hecke). Let f(z) = n=0 a nq n and g(z) = n=0 b nq n where q = e 2πiz and a n, b n are O(n ν ) for some real number ν. For positive integers k and N, the following conditions are equivalent (A) g(z) = ( i) k N k/2 z k f( 1/Nz). (B) Both Λ N (s, f) = (2π/ N) s Γ(s)L(s, f) and Λ N (s, g) = (2π/ N) s Γ(s)L(s, g) can be analytically continued to the whole s-plane, satisfy the functional equation (1.5.24) Λ(s, f) = Λ(k s, g) and Λ N (s, f) + a 0 s + b 0 k s is holomorphic on the s-plane and bounded on any vertical strip. See [5, 4.3.5] for proof. Theorem 1.5.6 (Weil). Let N be a positive integer and χ be a Dirichlet character modulo N. Suppose a n, b n are sequences of complex numbers such that a n, b n = O(n ν ) for some real number ν. If D is positive integer number, relatively prime to N, and if µ is a primitive Dirichlet character modulo D, we consider the Dirichlet series L a (s, µ) = n=0 µ(n)a nn s and L b (s, µ) = n=0 µ(n)b nn s. Let Λ a (s, µ) = (2π) s Γ(s)L a (s, µ) and Λ b (s, µ) = (2π) s Γ(s)L b (s, µ). 17

Let S be a set of primes including those dividing N. Assuming that the conductor D of µ is either 1 or a prime D / S, Λ a (s, µ) and Λ b (s, µ) have analytic continuation to the whole s-plane, are bounded in every vertical strips, and satisfy the functional equation (1.5.25) Λ a (s, µ) = i k µ(n)χ(d) τ(µ)2 D (D2 N) s+ k 2 Λb (k s, µ) Then f(z) = n=0 a nq n is a modular form of level N with nebentypus χ i.e. f M k (N, χ). 1.6. Hecke operators and Euler product. It is sometimes convenient to express the automorphy condition (1.3.1) in terms of group action. For every matrix with positive determinant a b γ = GL c d 2 (R) +, we define the right action of γ on a function f : H C by the transformation rule ( ) az + b (1.6.1) (f k γ)(z) = det(γ) k/2 (cz + d) k f cz + d A straightforward calculation shows that (f k α) k β = f k (αβ). Thus this transformation rule is indeed a right action of GL + 2 (R) on the space of holomorphic functions on H. With this definition, a modular form of weight k with respect to a Fuschian group Γ is a holomorphic function f on H such that f k γ = f for all γ Γ and which satisfies a growth condition near the cusps. It also follows from this definition that the algebra of double cosets of Γ in GL + 2 (Q) acts naturally on the space M k (Γ) as well as S k (Γ). The construction of the algebra of double cosets of a congruence subgroup Γ in Σ = GL + 2 (Q) relies on the following property. Lemma 1.6.1. Let Γ be a congruence subgroup and α Σ. Then Γ αγα 1 is a subgroup of Γ with finite index. This lemma can be reformulated in the following way : Each double coset ΓαΓ in Σ is a finite union of right cosets or left cosets (1.6.2) ΓαΓ = i Γα i where i runs over a finite set of indexes. It is convenient to see double cosets as notion of relative position between two let cosets. We will say that the two cosets Γα 1 and Γα 2 are in position α if α 1 α2 1 ΓαΓ. Lemma 1.6.1 can be reformulated in an yet another way : Lemma 1.6.2. For each left coset Γα 1 and each double coset ΓαΓ, there are only finitely many left cosets Γα 2 so that Γα 1 and Γα 2 are in position ΓαΓ. Let H Γ denote the free abelian group generated by a basis indexed by the set of double cosets of Γ in Σ. We simple write T α H Γ for the element in this basis indexed by the doubl coset ΓαΓ. We define a multiplication on H Γ by the following rule (1.6.3) T α1 T α2 = α c α α 1,α 2 T α where c α α 1,α 2 is the number of left coset Γβ such that Γβ 1 and Γβ are in position α 1 and Γβ and Γβ 2 are in position α 2, here Γβ 1 and Γβ 2 are fixed cosets in position α. The finiteness of the numbers c α α 1,α 2 follows immediately from 1.6.2. 18

We define the action of ΓαΓ = i Γα i on M k (Γ) by the formula (1.6.4) f k T α = i f k α i. This formula defines an action of the Hecke algebra H Γ on the space of modular forms M k (Γ). This action preserves the subspace of cusp forms S k (Γ). We consider the Hecke algebra H Γ(1) with respect to the full modular group Γ(1) = SL 2 (Z) as subgroup of Σ = GL 2 (Q) +. The double cosets of Γ(1) be described explicitly by the theory of elementary divisors. Proposition 1.6.3. Each double coset of Γ(1) in GL 2 (Q) + contains a unique diagonal matrix of the form d1 0 (1.6.5) α(d 1, d 2 ) = 0 d 2 with d 1, d 2 Q + such that d 1 d 1 2 is an integer. Proof. This can be best explained in term of relative position between lattices. A lattice of Q 2 is a free abelian group of rank two contained in Q 2. The map α α 1 (Z 2 ) defines a bijection between the set of left coset Γ(1)α in Σ = GL + 2 (Q) into the set of lattices of Q 2. If L, L Q 2 are two lattices, the theorem of elementary divisors assert that there exist a basis {x 1, x 2 } of L such that {d 1 x 1, d 2 x 2 } is a basis of L where d 1, d 2 are well defined positive rational numbers such that d 1 d 1 2 is an integer. Proposition 1.6.4. The algebra H Γ(1) is commutative. Proof. The transposed matrix g g being an anti-homomorphism of Σ, it induces an anti-homomorphism on H Γ(1). On the other hand, as it fixed the diagonal matrices α(d 1, d 2 ), it induces identity on H Γ(1). This means that H Γ(1) is a commutative algebra. The Hecke operators are self-adjoint with respect to the Peterson inner product on S K (Γ(1)). Recall that this inner product is defined by the integral (1.6.6) f, g = f(z)ḡ(z)y k dxdy. y 2 Γ(1)\H Here the invariant dxdy/y 2 on H and defines a measure on the quotient Γ(1)\H, the expression f(z)ḡ(z)y k is also invariant under Γ(1) and defines a function on Γ\H. Since f, g are cusp forms, f(z)ḡ(z)y k tends to zero near the cusps and thus is bounded function on Γ\H. Now the Peterson inner product is well defined since Γ(1)\H has finite measure with respect to dxdy/y 2. Theorem 1.6.5. The action of H Γ(1) on S k (Γ(1)) can be simultaneously diagonalized. Proof. The Hecke operators are self-adjoint with respect to to the Peterson inner product. A commutative algebra of self-adjoint operators on a finite dimensional vector space can be simultaneously diagonalized. We will next single out a particular family of Hecke operators that are relevant to L- functions. For every n N, we define the Hecke operator T (n) H Γ(1) by the following 19

formula (1.6.7) T (n) = T α(d1,d 2 ). The corresponding union of double cosets (1.6.8) T (n) = d 1,d 2 N,d 2 d 1 d 1 d 2 =N d 1,d 2 N,d 2 d 1 d 1 d 2 =N is the set of integral matrices of determinant n Γ(1)α(d 1, d 2 )Γ(1) (1.6.9) T (n) = {α Mat 2 (Z) det(α) = n}. Proposition 1.6.6. Let f = m=1 a mq m be a cusp form of weight k for Γ(1). Let f k T (n) = m=1 b mq m be the Fourier development of T (n)f. We have (1.6.10) b m = n k 2 d k+1 a md. a In particular ad=n,a m (1.6.11) b 1 = n k 2 +1 a n. Proof. We can make explicit the action of T (n) on modular forms by decomposing T (n) in left cosets of Γ(1) a b (1.6.12) T (n) = Γ(1). 0 d Thus (1.6.13) (1.6.14) f k T (n) = ad=n = ad=n a,b,d N ad=n,0 b<d 0 b<d n k/2 d k ( ) az + b n k/2 d k f d m=1 a m e 2πimaz d 0 b<d e 2πimb d The term 0 b<d e 2πimb d vanishes unless d m in which case it is equal to d. By replacing m by dm in the above formula, we get the following expression (1.6.15) (1.6.16) from which we derive (1.6.10) f k T (n) = ad=n = n k/2 d k+1 m=1 ad=n,a m m=1 a md e 2πimaz n k/2 d k+1 a md e m a Corollary 1.6.7. The Fourier coefficient a n of a normalized eigenform f of weight k with respect to Γ(1) is the eigenvalue of the operator n k 2 1 T (n). 20

Theorem 1.6.8. Let f = n=1 a nq n be a cuspidal eigenform of weight k for Γ(1) normalized so that a 1 = 1. Then we have (1.6.17) T (n)f = n k 2 +1 a n f. In particular, we have the relation (1.6.18) a mn = a m a n for all relatively prime integers m, n N. Moreover, the Dirichlet series L(s, f) admits a development in Euler product (1.6.19) L(s, f) = (1 a p p s + p k 1 2s ) 1. n=1 a n n s = p Proof. Let c(n) denote the eigenvalue of T (n) with respect to the eigenvector f. We have b 1 = c(n)a 1 = c(n) since a 1 = 1 with our normalization. Now (1.6.17) follows from (1.6.11). The multiplicative relation (1.6.18) follows from the relation in the Hecke algebra T (mn) = T (m)t (n) that holds for (m, n) = 1. The multiplicative relation implies a development in product of the Dirichlet series (1.6.20) L(s, f) = ( a p rp rs ). The formula (1.6.21) n=1 a n n s = p r=0 a p rp rs = (1 a p p s + p k 1 2s ) 1 r=0 follows from similar formula in the Hecke algebra. 1.7. Old and new forms. The theory of Hecke operators and expansion L-function as an Euler product can be generalized to any congruence subgroup. It will take however a much more complicated form. As the theory of Hecke operators can be significantly streamlined with the introduction of the adèles and the interpretation of modular form as automorphic forms on an adèlic group, we will postpone discussion after the adèles being introduced. For the record, we will just state the result of the theory of new forms, due to Atkin and Lehner. The proof will be postponed. Let M, N N such that M N. For any character χ : (Z/MZ) C, we have a character (Z/NZ) C also denoted by χ obtained by composing with (Z/NZ) (Z/MZ). For any integer d such that dm N, there is a map [d] : S k (M, χ) S k (N, χ) that associate to a form f S k (M, χ) the form z f(dz) that belongs to S k (N, χ). Let denote S k (N, χ) old the subspace generated by the image of [d] for all integers d, M so that dm N. The orthogonal complement of S k (N, χ) old is denoted S k (N, χ) new. Theorem 1.7.1. (1) The space of new forms S k (N, χ) new admits a basis consisting of normalized eigenform for the operators T p associated to the primes p not dividing N. (2) The L-function associated to such a normalized eigenform has a complete expansion as a Euler product. 21

(3) If f, f S k (N, χ) new such that for every prime p not dividing N, there exists a p such that T p f = a p f and T p f = a p f then f and f are proportional. 2. Representations of GL(2, R) + 2.1. Representations of locally compact groups. Let G be a topological group that is locally compact. We will be interested in representations of G on Hilbert spaces. Definition 2.1.1. A representation of G on a Hilbert space H is a homomorphism π from G to the group of continuous linear transformations of H so that for evry v H, the map g π(g)v is continuous. If moreover π preserves the inner product on H, we will say that the representation π is unitary. Lemma 2.1.2. Every representation is locally bounded. In other words, for every compact K of G, there exists a positive real number C such that π(g) < C for every g K. Proof. For every v, the vectors π(g)v with g K form a compact set of H which is necessarily bounded. The lemma follows from the uniform boundedness principle. Let C c (G) denote the space of complex valued continuous functions on G with compact support. Assume that G is unimodular. The Haar measure dg defines a positive linear form C c (G) C φ φ(g)dg. G The Haar measure also provides C c (G) with a structure of algebra under the convolution product (2.1.1) φ ψ(x) = φ(xy 1 )ψ(y)dy. G Let π be a representation of a locally compact topological group G on a Hilbert space g. We consider the integral π(φ)v = φ(g)π(g)vdg G for every φ C c (G) and v H. The above integral can be defined as follows. We first consider the continous linear form v φ(g) π(g)v, v dg. G By Riesz representation theorem there exists a unique vector π(φ)v H so that π(φ)v, v = φ(g) π(g)v, v dg. This defines an action of C c (G) on H. G Definition 2.1.3. A sequence of positive functions φ n is said to be approximating the delta function of the identity of G in the following sense (1) φ n C c (G) are supported in a certain compact K of G (2) G φ n(g)dg = 1 for every n (3) for every neighborhood U of 1 G we have lim n G U φ n(g)dg = 0. 22

Lemma 2.1.4. Let π be a representation of G on a Hilbert space H. If φ n is a sequence of function approximating the delta function of 1 G, then we have for all v H. lim π(φ n)v = v n Proof. Let v V. For every ɛ > 0, there exists a neighborhood U of 1 G so that for every g U, π(g)v v < ɛ. For an integer n large enough, we have lim n G U φ n(g)v dg < ɛ since the family φ n is bounded and lim n G U φ n(g)dg = 0. We can also check that π(φ U n)v v < ɛ for n large enough. 2.2. Representations of the circle group. The theory of unitary representations of the circle group is very much a reformulation of the series of Fourier series associated to square integrable functions on the circle T = R/Z. The characters of is of the forms χ n (x) = e 2πinx. Let H = L 2 (T) denote the space of square integrable functions on T. The translation by T defines a unitary representation of T on H. Theorem 2.2.1. Let π be a unitary representation of T on a Hilbert space H. For every v H and n Z, we consider v n = π(χ n )v = e 2πinx π(x)vdx. Let H n denote the image p n : v v n = π(χ n )v. (1) For n m, the subspaces H n and H m are orthorgonal. (2) The space H is the Hilbert direct sum of its subspace H n. Proof. Only the last statement is non trivial. For the last statement, it suffices to prove that for all v H, v = lim N N n= N p n(v). Since v n=n n= N p n(v) is orthogonal to N n= N H n, for every v N n= N H n, we have v v v T n=n n= N p n (v). It follows that we only need to construct a sequence v N N n= N H n so that lim v N v N = 0. This can be achieved by constructing a delta sequence of functions φ N on T so that for all N, φ N is a linear combination of e 2πinx with n N. Classical examples of this is Fejer s kernel (2.2.1) (2.2.2) K N (x) = = N n= N 1 N + 1 (1 n N + 1 )e2πinx 23 sin 2 (π(n + 1)x) sin 2 (πx)

On the complement of neighborhood of 0 T, sin 2 (πx) > α for some given positive number α which implies that K N (x) < e 1 /(N + 1) for all N. This implies that Fejer s sequence approximate the delta function. A vector v of some finite direct sum N n= N H n is called a finite vector i.e. its transforms π(x)v with x T generates a finite dimensional vector space. It follows from the above theorem that there are non zero finite vectors in any unitary representation of the circle groups, and moreover, the finite vectors form a dense subspace of the Hilbert space. This statement can be generalized to arbitrary compact group. Lemma 2.2.2. Let (π, H) be a Hilbert representation of a compact group K.Then there exists a hermitian inner product on H inducing the same topology as the original one so that π is unitary i.e. π(g)v, π(g)v = v, v for all g K and v, v H. Proof. [2, 2.4.3] Let v, v 1 denote the given inner product on H. For all v, the function k π(k)v, π(k)v 1 is a continuous function onthe compact group K which is then necessarily bounded. By the uniform boundedness principle, there exist a real number C > 0 such that π(k)v < C v for all non zero vector v H. This also implies that π(k)v > C 1 v. The inner form v, v = π(k)v, π(k)v 1 dk K is obviously positively definite. The inequalities C 1 v < π(k)v < C v imply that it defines the same topology on H as v, v 1. 2.3. Lie groups. Let G be a real Lie group. Let g denote its Lie algebra. Proposition 2.3.1. There exists natural isomorphism between (1) The tangent space of G at the origin g. (2) The space of left invariant vector fields on G. (3) The space of left invariant derivations of C (G). For every vector X g, there is a unique invariant vector field L X having X as the vector X at the origin. For every g G, the left translation map l g : G G induces an isomorphism dl g : g T g G and we have L X,g = dl g (X). Left invariant vector fields are equivalent to left invariant derivations. We have the following formula for the bracket L [X,Y ] f = (L X L Y L Y L X )f. Proposition 2.3.2. There exists a map exp : g G that is a local homeomorphism in a neighborhood of the origin in g such that, for any X g, t exp(tx) in an integral curve for the left invariant vector field L X. Moreover exp((t + u)x) = exp(tx) exp(ux). For every smooth function f C (G), we have the following formula for the left invariant derivation (L X f)(g) = d dt f(g exp(tx)) t=0. Definition 2.3.3. A representation of the Lie algebra is a linear application π : g End(V ) from g to the space of endomorphisms of a vector space V such that for all X, Y g, we have π[x, Y ] = π(x)π(y ) π(y )π(x). 24