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AUTOMORPHIC FORMS ON GL2 This is an introductory course to modular forms, automorphic forms and automorphic representations. (1) Modular forms (2) Representations of GL2(R) (3) Automorphic forms on GL2(R) (4) Ad`elesand id`eles (5) Representations of GL2(Qp) (6) Automorphic representations of GL2(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 = SL2(R), K = SO2(R), H = G=H is the upper half- ¯ plane, D is the open unit disc. Γ will denote a discrete subgroup of SL2(R), Γ its image in ¯ PGL2(R). In particular, Γ(1) = SL2(Z) and Γ(1) is its image in PGL2(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 GL2 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 a b az + b (1.1.1) z = c d cz + d if z denotes the standard coordinate of P1. This formula is valid for any coefficients fields. 1 1 In particular, GL2(R) acts on P (R) and GL2(C) acts compatibly on P (C). It follows that GL2(R) acts on the complement of the real projective line inside the complex projective line 1 1 − P (C) − P (R) = H [ H where H (resp. H−) is the half-plane of complex number with positive (resp. negative) + imaginary part. Let GL2 (R) denote the subgroup of GL2(R) of matrices with positive determinant; it is also the neutral component of GL2(R) with respect to the real topology. + 1 1 − Since GL2 (R) is connected, its action on P (C) − P (R) preserves H and H . Of course, the above assertion is a consequence of the formula az + b ad − bc (1.1.2) = = =(z): cz + d jcz + dj2 1 which derives from a rather straightforward calculation az + b (az + b)(cz¯ + d) = cz + d jcz + dj2 bd + aczz¯ + bc(z +z ¯) + (ad − bc)z = jcz + dj2: This equation becomes even simpler when we restrict to the subgroup SL2(R) of real coefficients matrix with determinant one az + b =(z) (1.1.3) = = : cz + d jcz + dj2 From now on in this chapter, we will set G = SL2(R). Lemma 1.1.1. The group G acts simply transitively on the upper half-plane H. The isotropy group of the point i 2 H is the subgroup K = SO2(R) of rotations : cos θ sin θ (1.1.4) k = θ − sin θ cos θ Proof. The equation ai + b = i ci + d implies that a = d, b = −c in which case the determinant condition ad − bc = 1 becomes a2 + b2 = 1. Thus the matrix is of the form (1.1.4). Let z = x + iy with x 2 R and y 2 R+. It is enough to prove that there exists a; b; c; d 2 R with ad − bc = 1 such that ai + b = z: ci + d 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 = y1=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 dx2 + dy2 (1.1.5) ds2 = y2 on H, as well as the density µ = dxdy=y2 is invariant under the action of G. a b Proof. With the notations γ = and z0 = γz, we have c d (ad − bc) (1.1.6) dz0 = dz: (cz + d)2 This calculation has the following concrete meaning. The smooth application g : H ! 0 H maps z 7! z . It induces linear application on tangent spaces TzH ! Tz0 H and its ∗ ∗ ∗ ∗ dual linear application Tz0 H ! TzH. The cotangent space Tz0 H (resp. TzH) is a one- dimensional C-vector space generated by dz0 (resp. dz). The linear application sends dz on ((ad − bc)=(cz + d)2)dz. 2 2 2 The element dz induces the canonical quadratic form dx + dy on TzH viewed as 2- 02 02 dimensional real vector space. Similarly, we have the quadratic form dx + dy on Tz0 H. The equation (1.1.6) implies that (cz + d)2 dx02 + dy02 = (dx2 + dy2): jcz + dj4 It follows that the metric ds2 = (dx2 + dy2)=y2 is invariant under G, according to (1.1.2). 2 The same argument applies to the density µ = dxdy=y . Lemma 1.1.3. The Cayley transform 1 −i z − i (1.1.7) z 7! cz = z = : 1 i z + i maps isomorphically H onto the unit disk D = fz 2 C j jzj < 1g. The inverse transformation is 1 1 1 i(1 + w) (1.1.8) w 7! c−1w = w = : 2 i −i 1 − w The metric ds2 = (dx2 + dy2)=y2 on H transports on the metric 4(du2 + dv2) (1.1.9) d s2 = D (1 − jwj2)2 where w = u + iv. We also have 4dudv (1.1.10) dxdy=y2 = : (1 − jwj2)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 2 H, we have jz − ij < jz + ij so that jc(z)j < 1. It follows that c(H) ⊂ D. For every w 2 D, the straightforward calculation i(1 + w) −2=(w) + i(1 − jwj2) (1.1.11) = 1 − w j1 − wj2 shows 1 − jwj2 (1.1.12) y = > 0: j1 − wj2 if z = c−1w and y = =(z). It follows that c−1(D) ⊂ H. By using the chain rule we have 2idw dz = : (1 − w)2 If we write w = u + iv in cartesian coordinates, then we have 4(du2 + dv2) dx2 + dy2 = : j1 − wj4 It follows that dx2 + dy2 4(du2 + dv2) = : y2 (1 − jwj2)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 Z 1 2(1 − jφ(t)j2)−1p(dx(t)=dt)2 + (dy(t)=dt)2dt 0 that is at least Z 1 Z a 2 −1 2dt 2(1 − x(t) ) jdx(t)=dtjdt = 2 0 0 (1 − t ) The shortest curve joining 0 and a is thus a part of a radius in the unit disc. For every two points x0; x1, there is g 2 SL2(R) that maps H on D, cg(x0) = 0 and cg(x1) = a where a is a positive real number satisfying a < 1. Here c : H ! D is the Cayley transform. The geodesic joining x0 with x1 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 transforma- tion (1.1.1). Prove that the subgroup that map the unit disc D onto itself is a b (1.1.13) SU(1; 1) = j jaj2 − jbj2 = 1 : ¯b a¯ 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 2 D is the rotation group eiθ 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 SL2(Z) and its subgroup of finite indices. We will call Fuchsian group a discrete subgroup of SL2(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) 7! (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 fγ 2 ΓjγU \ V 6= ;g is a finite. Because the group G = SL2(R) acts on H with compact stabilizer, the subset fg 2 GjγU \ V 6= ;g is compact. Its intersection with the discrete subgroup Γ is finite. Corollary 1.2.2.
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