AUTOMORPHIC FORMS ON GL 2

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1 AUTOMORPHIC FORMS ON GL 2 Introduction This is an introductory course to modular forms, automorphic forms and automorphic representations. We will follow the plan outlined in a book of Bump [2] but also use materials from other sources as well. (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 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) 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 can also be justified by a direct computation : az + b ad bc (1.1.2) I = cz + d cz + d I(z). 2 1

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 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) 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 a rotation matrix (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 The metric (1.1.5) ds 2 = dx2 + dy 2 on H, as well as the associated volume form dv(z) = dxdy/y 2 is invariant under the action of G. a b Proof. If γ = with ad bc = 1, we have c d (1.1.6) d(γz) = y 2 dz (cz + d) 2 It follows that the metric dx 2 +dy 2 is transformed with by the factor cz +d 4 and the metric ds 2 = dx 2 + dy 2 /y 2 is invariant. Lemma 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 i i w + 1 w = i ( w + 1). 2

3 The metric ds 2 on H transports on the metric (1.1.9) d D s 2 = 4(dx2 + dy 2 ) (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, a direct computation shows that I(c 1 w) = 1 w w 2 > 0. It follows that c 1 (D) H. Using the chain rule we have i dz = ( w + 1) dw. 2 It follows that the metric ds 2 = (dx 2 + dy 2 )/y 2 on H corresponds to the metric d D s 2 = 4(dx 2 + dy 2 )/(1 w 2 ) 2 on D. Lemma 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 1 0 2(1 φ(t) 2 ) 1 (dx(t)/dt) 2 + (dy(t)/dt) 2 dt that is at least 1 2(1 x(t) 2 ) 1 dx(t)/dt dt = 0 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 [2, Ex ] 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 (1.1.10) 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. 3

4 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 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 For every Fuchsian group Γ, the quotient Γ\H is a Haussdorf topological space. 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) cos(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 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 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 d. 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 d can be given a complex structure with uniformizing parameter w d where w 4

5 is the standard uniformizing parameter of D around 0. This defines a complex structure on a neighborhood of [z] in Γ\H. Definition 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 Γ\H is a Haussdorff space. 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 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 A discrete subgroup Γ of SL 2 (R) is called a Fuchsian group of first kind if X Γ = Γ\H is compact. Proposition If X Γ is compact, then the numbers of elliptic points and cusps of Γ in Γ\H are finite. Theorem (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 5

6 (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 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 = is a fundamental domain of Γ. γ Γ Z 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 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) (1.2.9) T = and S = They act on H by the following rules T z = z + 1 and Sz = 1/z. Lemma 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. 6 F γ

7 (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. 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 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 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 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 7

8 number of Γ-equivalence classes of cusps. Then the genus of Γ\H is Proof. See [7, 1.40]. The map g = 1 + µ 12 m 2 4 m 3 3 m 2. 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 + P (e P 1) 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 over is µ m. By summing altogether, we get the desired formula for the genus of Γ\H. Corollary If Γ do not have elliptic points then the genus of Γ\H is g = 1 + µ 12 m 2. Exercice [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 and (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) 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 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. 8

9 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. Lemma 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 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 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 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 : 9

10 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. 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 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 ). 10

11 In particular dim S 2 (Γ) = g (calculated in ) 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 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}. Corollary 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 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 The rational function j : G 3 4/ defines an isomorphism from X Γ(1) onto P 1 C 11

12 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 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. 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 Let k be an integer greater or equal to two. We have the formula (1.4.2) for all z H. n Z (n z) k = (2πi)k (k 1)! n=1 n k 1 e 2πinz 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 C

13 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 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 valid for k > 2. σ k 1 (n) n k 1 = d n n N 1 ζ(k 1) dk 1 Proposition 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 where defines a continuous function on the fundamental domain of Γ. Assume now Γ = SL 2 (Z). Since q = e 2πy, the vanishing of the constant terms of the Fourier expansion of f by 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 imporved according to the Ramanujan-Peterson conjecture. Theorem 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 τ(p) 2p 11/2 13

14 that is the original conjecture of Ramanujan 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 and 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 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). 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) f(iy)y s dy 0 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 (1.5.6) (1.5.7) (1.5.8) (1.5.9) 0 f(iy)y s dy y = = 0 1 a n e 2nπy y s dy y 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 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). 14

15 Appendix A. Review on compact Riemann surfaces Let X be a compact Riemann surface. By GAGA theorem, X is a smooth projective curve. Concretely, this means that the field of meromorphic functions F on X is finite extension of the field of fractions C(t) of the polynomial ring C[t]. In fact any non constant meromorphic function f F defines a finite morphism f : X P 1 C and by assigning t f, we make F a finite extension of C(t). A.1. Divisors. The group of divisors Div(X) is the free abelian group with basis the set of points x X. Its elements are finite linear combinations i d ix i with d i Z and x X. The application i d ix i i d i defines a homomorphism deg : Div(X) Z. We denote Div 0 (X) the group of divisors of degree 0. A divisor D Div(X) is said to be effective if D = i d ix i with d i 0. We will write simply D 0 if D is effective. For every f F, we have a divisor div(f) = x ν x(f)x where ν x (f) is the vanishing or pole order of f at the point x. If u x is a parameter of X at x then f u νx x up to the multiplication by invertible function at x. We have div(f) Div 0 (X). For every Zariski open subset U X, we also the group Div(U) and the notion of efective divisors of U. For every meromorphic function f F, we also have a divisor div U (f) = x U ν x(f)x. We will denote O X the structural sheaf of the the algebraic curve X whose generic fiber is F. For every Zariski open subset U X, the sections of O X (U) are regular algebraic functions on U. In other words Γ(U, O X ) = {f F div U (f) 0}. For every x, the local ring of germs of regular functions at x is O X,x = {f F ν x (f) 0}. This is a regular local ring of dimension one i.e. its maximal ideal is generated by one element. A generator of the maximal ideal of O X,x is called a parameter of X at x. More generally, for every D Div(X), the sheaf O X (D) is defined by Γ(U, O X (D)) = {f F D U + div U (f) 0}. For every x, the group of germs of its sections regular at x is O X (D) x = {f F d x + ν x (f) 0}. Its is clear that for every U X, H 0 (U, O X (D)) is a H 0 (U, O X )-module and for every x X, O X (D) x is a O X,x -free module of rank one. This means that for every D Div(X), O X (D) is a locally free O X -module of rank one, in other words a line bundle over X. The following statement is a consequence of the Riemann-Roch theorem and the duality of Serre that we will recall later. Proposition A.1.1. For every D Div(X) the vector space Γ(X, O X (D)) = {f F D + div(f) 0} is a finite dimensional. If deg(d) < 0, we have Γ(X, O X (D)) = 0. If deg(d) > 2g 2 where g is the genus of X, we have dim Γ(X, O X (D)) = 1 g + deg(d). 15

16 A.2. Line bundles. This theorem generalizes to any line bundle because every line bundle L is of the form O X (D). More precisely, let a L OX F be a meromorphic section of L. For every x X, let ν x (a) be the integer so that we have the equivalence relation a l x u νx(a) x up to multiplication by an invertible unction at x. Here l x is a generator of L x as O X,x -module and u x is a generator of the maximal ideal of O x. Let define D = div(a) = x X ν x(a)x. Then we have a canonical isomorphism L = O X (D). Over the generic fiber, this is the isomorphism between one-dimensional F -vector spaces F L OX F assigning 1 a. Even if any line bundle L on X is of the form L = O X (D) for some divisor D, line bundle is not naturally equipped with a meromorphic section so the line bundles and divisors are not equivalent notions. Nevertheless the integer deg(div(a)) does not depend on the choice of the meromorphic section a of L since two different meromorphic section differ by a meromorphic function. Therefore deg(l) = deg(div(a)) is well defined. Theorem A.2.1 (Riemann-Roch). The cohomology groups H i (X, L) are finite dimensional and vanish if i / {0, 1}. We have dim H 0 (X, L) dim H 1 (X, L) = 1 g + deg(l). The sheaf of 1-forms Ω X/k of X over k is a line bundle over X of degree 2g 2. For every affine open subset U X with ring of regular functions A = Γ(U, O X ), we have Γ(U, Ω X/k ) = Ω A/k where Ω A/k is the A-module of Kahler differentials. Theorem A.2.2 (Serre). There is a canonical non degenerate pairing between H 0 (X, L) and H 1 (X, L 1 Ω X/k ). In particular, we have a canonical isomorphism H 1 (X, Ω X/k ) C. A.3. Covering. Let us now review the Hurwitz theorem. Let f : Y X be a finite morphism between smooth projective curves over C. Pick a point y Y with image x X, and let u x be a local parameter of X at x and u y a local parameter of Y at y. We say that f is étale at y if u x is a local parameter of Y at y, in other words u x and u y differ by an invertible function at y. Since our base fields in C, this happens for all but finitely many points y Y. A non étale point y Y is also called a ramified point. There exists an integer e y, the ramification index, such that u x u ey y up to an invertible function. Theorem A.3.1 (Hurwitz). Let f : X Y be a finite morphism of degree d between smooth projective curves over C. We have the relation 2g Y 2 = d(2g X 2) + y where we sum over all ramification points of Y. (e y 1) One can pull back a 1-form from X to Y. This defines homomorphism f Ω X Ω Y which which is an injective map whose cokernel is a torsion sheaf Ω Y/X supported by the ramified points. We have deg(f Ω X ) = d(2g X 2) and deg(ω Y ) = 2g Y 2. It is not difficult to evaluate the length of the cokernel. Let y Y be a ramified point. We derives from the relation u x u ey y that du x u ey 1 y du y which means that the length of the direct factor of Ω Y/X supported by y is e y 1. The Hurwitz theorem follows. A.4. Adelic desctiption. Following Weil, any line bundle admit adelic description as follows. Recall that F x is the completion of F with respect to the topology defined by the maximal ideal of the local ring O X,x and O x its ring of integers. In constrast of the local 16

17 ring O X,x remembers about the curve X because its field of fractions is F, the completed local ring O x is always isomorphic to the ring of formal series of one variable. Let A F denote the ring of tuples (f x ) x X with f x F x for all x X and f x O x for all but finitely many x X. In particular L A contains both F and x O x. We can attach to line bundle L on X the following adelic data. First we have L = L OX F is a one-dimensional F -vector space. At each point x X, the one-dimensional vector space L x = L F F x is equipped with a O x -submodule L x = L OX O x which is a free O x -module of rank one. The adelic data (L, (L x ) x X ) is required to satisfy the following condition : for every nonzero element a L, a is generator of L x for all but finitely many x X. We denote L A = L F A F that contains both L and x L x. We observe that the adelization of line bundle has an obvious generalization to vector bundles. Proposition A.4.1. The cohomology groups are given by the following formula (A.4.1) H 0 (X, L) = L x L x (A.4.2) H 1 (X, L) = L A /(L + x L x ). For each x X, we have a canonical map Ω X OX F x C given by the residue at x. By taking the sum of residue, we have a map L A C whose restriction to x Ω X OX O x vanish by construction, and whose restriction to Ω X,x OX F vanishes by the residue theorem. This defines a canonical map H 1 (X, L) C that appears in Serre s theorem. Appendix B. Fourier series Appendix C. Fourier transform and the Poisson summation formula See [9, chapter 5] for more details and proofs. C.1. Schwartz functions and the Fourier transform. A Schwartz function on R is a smooth (indefinitely differentiable) functions f : R C so that f along with all its derivatives f, f (2),... are rapidly decreasing, in the sense that (C.1.1) sup x k f (l) (x) < for every k, l 0 x R We denote S(R) the space of all Schwartz functions. Smooth functions with compact support are also Schwartz functions. Simple example of Schwartz functions are P (x)e x2 where P (x) is a polynomial function. The main property of the Schwartz class of function is its stability under the Fourier transform f ˆf with ˆf(y) = e 2πixy f(x)dx. Theorem C.1.1. If f S(R) then ˆf S(R). Proof. It can be easily checked that the Fourier transform of a Schwartz function is bounded. The theorem derives from the fact that the Fourier transform exchanges diffentation and multiplication l d y k ˆf(y) dy 17

18 is the Fourier transform of 1 (2πi) k which is also a Schwartz function. C.2. The Poisson summation formula. k d ( 2πix) l f(x) dx Theorem C.2.1. Let f S(R) be a Schwartz function. The series f(n) and ˆf(n). n= n= are absolutely convergent. Moreover, we have the equality f(n) = f(n). Proof. [9, p.154]. n= n= It is sometimes useful to relax the rapidly decreasing condition. A function f is said to be a sufficiently decreasing condition if there exist constant ɛ > 0 and A > 0 such that A f(x) < 1 + x. 1+ɛ The series n= n= f(n) is absolutely convergent as long as the function f is sufficiently decreasing. In fact the Poisson summation formula and the proof [9, p.154] holds if both f and ˆf are sufficiently decreasing. Proposition C.2.2. For any function f that is twice continuously derivable and if f, f, f (2) are sufficiently decreasing, the Poisson summation formula holds. References [1] Borel, Automorphic forms on SL 2 (R) [2] Bump, Automorphic forms and representations. [3] Kudla, Tate s thesis [4] Lang, SL 2 (R) [5] Miyake, Modular forms [6] Serre, Cours d arithmétique [7] Shimura, Introduction to the arithmetic theory of automorphic forms [8] Reed, Simon, Functional analysis [9] Stein, Shakarchi, Fourier analysis: an introduction [10] Stein, Shakarchi, Complex analysis [11] Wallach, Real reductive groups I. [12] Warner, Harmonic analysis on semisimple Lie groups I. 18