AUTOMORPHIC FORMS on GL This Is an Introductory Course to Modular

Home , Automorphic form, Automorphic function, Matrix coefficient

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 |cz + d|2 1 which derives from a rather straightforward calculation az + b (az + b)(cz¯ + d) = cz + d |cz + d|2 bd + aczz¯ + bc(z +z ¯) + (ad − bc)z = |cz + d|2.

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 |cz + d|2

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 ∈ 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 ∈ 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 = 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). |cz + d|4 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 = {z ∈ C | |z| < 1}. 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 − |w|2)2 where w = u + iv. We also have 4dudv (1.1.10) dxdy/y2 = . (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 i(1 + w) −2=(w) + i(1 − |w|2) (1.1.11) = 1 − w |1 − w|2 shows 1 − |w|2 (1.1.12) y = > 0. |1 − w|2 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 = . |1 − w|4 It follows that dx2 + dy2 4(du2 + dv2) = . y2 (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 Z 1 2(1 − |φ(t)|2)−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) ) |dx(t)/dt|dt = 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 ∈ 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 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|2 − |b|2 = 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 ∈ 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 {γ ∈ Γ|γU ∩ V 6= ∅} is a finite. Because the group G = SL2(R) acts on H with compact stabilizer, the subset {g ∈ G|γU ∩ V 6= ∅} 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 ∈ SL2(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 1 Any element g ∈ SL2(C) has at least on fixed point in P (C). If g ∈ SL2(R) is not elliptic, it must have fixed points on P1(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 x (1.2.1) 0 with ∈ {±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 Γx of x in Γ is a finite group because it is the intersection of a compact group with a discrete group. In fact, since SO2(R) is isomorphic to the circle, for every x ∈ H,Γx is a finite cyclic group. If Γx 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 Γz = 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 Γx = Γ ∩ Z such that Γz is a finite group larger than Γ ∩ Z.Γz is then a finite cyclic group µk. 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 d the action of µk can be given a complex structure with uniformizing parameter w 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 ∈ P1(R) is called a cusp for Γ if it is the fixed point of a nontrivial parabolic element.

In that case Γx is isomorphic to the product of Z with an inﬁnite cyclic group, the ﬁrst factor Z = µ2 being the center of G. Let PΓ denote the set of cusps of Γ. We set ∗ H = H ∪ PΓ. We consider the topology on H∗ by adding to the real topology on H a family of neighborhoods of each cusp x ∈ PΓ. If x = ∞, we take the family ∗ (1.2.3) Ul = Ul ∪ {∞} with Ul = {z ∈ H | =(z) > l}

The family of neighborhoods of other cusps are constructed from the Ul 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 = { |a ∈ ×, b ∈ } ∞ 0 a−1 R R

By definition, ∞ is a cusp of Γ if Z(Γ ∩ G∞) is a subgroup of the form mn (1.2.5) { | ∈ {±1}, m ∈ } 0 Z 2iπz/n ∗ for some fixed integer n. The map z 7→ e 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 SL2(R) is called a Fuchsian group of ﬁrst kind ∗ if XΓ = Γ\H is compact.

Proposition 1.2.8. If XΓ 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 SL2(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 S (i) H = γ∈Γ γF ; (ii) if U is the set of interior points of F then F = U¯ ; (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 z0 ∈ H which is not fixed by any element γ ∈ Γ − Z. For every γ ∈ Γ, we put

Fγ = {z ∈ H | d(z, z0) ≤ d(z, γz0)}

Uγ = {z ∈ H | d(z, z0) < d(z, γz0)}

Cγ = {z ∈ H | d(z, z0) = d(z, γz0)} Here, d indicates the hyperbolic distance on H deﬁned by the metric ds2 = (dx2 + dy2)/y2. The intersection \ F = Fγ γ∈Γ−Z 6 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.

1 Lemma 1.2.11. Let γ ∈ SL2(Q) act on P (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 SL2(Z) and its subgroups of congruence

(1.2.6) Γ(N) = {γ ∈ SL(2, Z) | γ ≡ 1 mod N} 1 ∗ (1.2.7) Γ (N) = {γ ∈ SL(2, ) | γ ≡ mod N} 1 Z 0 1 ∗ ∗ (1.2.8) Γ (N) = {γ ∈ SL(2, ) | γ ≡ mod N} 0 Z 0 ∗

In particular, we will use the convenient notation Γ(1) = SL2(Z) for the full modular group. We consider the case of the full modular group. Let F denote the domain defined by the conditions |<(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 Γ0 denote the subgroup of Γ(1) generated by the transformations S and T as above. (1) For every z ∈ H, there exists γ ∈ Γ0 such that γz ∈ F . (2) If z, z0 ∈ F and γ ∈ Γ(1) non trivial such that γz = z0 then z, z0 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 z0 = γz conjugate to z such that =(z0) ≥ =(z). We can assume that =(z) is maximal among all the conjugates γz with γ ∈ Γ0. With the help of the translation T , we can assume that z belongs the the vertical strip <(z) ≤ 1/2. We only need to prove that under these assumptions, we have |z| ≥ 1. If |z| < 1, we would have =(−1/z) > =(z) that would contradict the maximality of =(z). It follows that z ∈ F . Let z, z0 ∈ F and a b γ = ∈ Γ(1) such that z0 = γz. c d 7 We can assume that =(z) ≤ =(z0). 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 z0 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 γ0 ∈ Γ0 such that γ0γz ∈ F . The assumption that z lie in the interior of F implies γ0γ = 1, thus γ ∈ Γ0. 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). ∗ 1 The quotient XΓ(1) = Γ(1)\H is isomorphic to P (C) as complex analytic space. Up to equivalence, Γ(1)¯ has one elliptic point of order 2 that is i ∈ H with b2 = −1 and one elliptic point of order 3 that is j ∈ H with j3 = 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 1 Riemann surface that is homeomorphic to the sphere. It is isomorphic to P (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 xn be a sequence of points of Γ\H . We need to prove that there exists ∗ a convergent subsequence. Let zn be a sequence in H so that xn = Γzn is the image of zn in ∗ ∗ Γ\H. Letx ¯i denote the image of zn in Γ(1)\H . Since Γ(1)\H is compact, we can assume ∗ ∗ thatx ¯n converges thex ¯ ∈ Γ(1)\H . Let z ∈ H be a preimage ofx ¯. There exists γn ∈ Γ(1) so that γnzn converges to z. Because Γ/Γ(1) is finite, after extracting a subsequence, we can assume that there exist γ ∈ Γ(1) so that γn ∈ γΓ(1) for all n. It follows that xn converges to γ−1x 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 m2, m3 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 µ m m m g = 1 + − 2 − 3 − ∞ . 12 4 3 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 X 2g − 2 = −2µ + (eP − 1) P with P in the set of ramified points, eP being the index of ramification. Summing eP over the ramified points P over j we get 2(µ − m3)/3. The same sum over i is (µ − m2)/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 µ m g = 1 + − ∞ . 12 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 P1(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)kf(z) cz + d for all z ∈ H and a b ∈ SL(2, ) c d Z 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 = e2πiz. The equation 1.3.1 implies in particular f(z + 1) = f(z), and thus f has a Fourier expansion

X 2iπnz X n (1.3.2) f(z) = ane = anq . n∈Z n∈Z

The function f is holomorphic at the cusp ∞ if in the above expansion an = 0 for n < 0. If furthermore a0 = 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 PGL2(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 n f admits a Fourier expansion f =n∈Z anq with an = 0 for n < 0 with respect to the variable q = e2iπz/m. 9 Lemma 1.3.1. Suppose that ∞ is a cusp of a Fuschian group Γ of first kind with Γ∞ P n generated by the matrix (1.3.3). Let f be a modular form of weight k and let anq with 2iπz/m P n q = e denote its Taylor expansion near this cusp. Then the series anq converges absolutely and uniformly on every compact in H. 2iπz/m Proof. The function z 7→ q = e defines an isomorphism between Γ∞\H and the punc- tured disc D − {0}. By assumption modular form f defines a holomorphic function on P∞ n D − {0} that extends holomorphically to D. This implies that the Taylor series n=0 anq converges absolutely uniformly on every compact contained in D.

Deﬁnition 1.3.2. Let Γ be a Fuschian group of ﬁrst kind. We denote Mk(Γ) the space of modular forms of weight k for Γ. We denote by Sk(Γ) the space of cusp forms of weight k for Γ. We also denote Ak(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, A0(Γ) is the ﬁeld FΓ of meromorphic functions on XΓ. We have M0(Γ) = C and S0(Γ) = 0.

We will now determine the dimension of Mk(Γ) and Sk(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 A2k(Γ) of meromorphic automorphic forms of weight 2k and the space Ω⊗k ⊗ F of meromorphic k-fold XΓ OXΓ Γ differential form on XΓ. In particular, A2k(Γ) is a one-dimensional FΓ-vector space. ⊗k Proof. For every f ∈ Ak(Γ), the k-fold differential form f(z)(dz) is Γ-invariant. It descends to a meromorphic k-fold differential form ωf on Γ\H. The condition of meromorphicity of f at the cusps impies that ωf is a meromorphic form on XΓ. The application f 7→ ωf induces an isomorphism A (Γ) → Ω⊗k ⊗ F . 2k XΓ OXΓ Γ

In order to calculate the dimension of Mk(Γ), we will express the condition of holomorphicity of f in terms of the zero divisor of ωf on XΓ 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 ∈ XΓ. The function f is holomorphic at z0 if and only if ωf is holomorphic at x. • Let z ∈ H be an elliptic point of index e and let denote x ∈ XΓ its image. Let t e be a local parameter at z ∈ H and u a local parameter at x ∈ XΓ. We have t ∼ u where the equivalence means equal up to an invertible function on a neighborhood of z. By derivation, we have du ∼ ze−1dz. Raising to the power k, we have (dz)⊗k ∼ −k(e−1) ⊗k z (du) . Let denote νx(f) the valuation of f with respect to the parameter u i.e. νx(u) = 1 and νx(t) = 1/e. Let us calculate the order of vanishing of the k -fold diﬀerential form ⊗k −k(e−1) ⊗k ωf = f(dz) ∼ fz (du) at x. We have ordx(ωf ) = νx(f) − k(1 − 1/e).

The function f is holomorphic at z if and only if νx(f) ≥ 0 which is equivalent to

ordx(ωf ) + k(1 − 1/e) ≥ 0. 10 Since ordx(ωf ) is an integer, the above inequality is equivalent to

ordx(ωf ) + [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 ordx(ωf ) ≥ 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\XΓ. The development of f at the cusp has the form

X n f(z) = anq n∈Z where q = 2iπmz for some positive integer m. Let r be the least integer such that ⊗k ar 6= 0. We note νx(f) = r. Let us denote ωf = fdz . Since dz ∼ dq/q we have −k ⊗k ωf ∼ fq dq . By construction, q is a local parameter of XΓ at x. It follows that

νx(ωf ) = νx(f) − k.

Thus f is holomorphic at the cusp ∞ i.e. νx(f) ≥ 0 if and only if

νx(ωf ) + k ≥ 0

and f vanishes at the cusp ∞ i.e. νx(f) ≥ 1 if and only if

νx(ωf ) + k ≥ 1.

In wight two case k = 1, f is holomorphic at ∞ if ωf is a logarithmic one form and f is a cusp form if and only if ωf 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 M2(Γ) is canonically 0 isomorphic with the space H (XΓ, ΩXΓ (cusp)) of one form with logarithmic singularities at the cusps. The space S2(Γ) is canonically isomorphic with the space of holomorphic one form of XΓ 0 S2(Γ) = H (XΓ, ΩXΓ ).

In particular dim S2(Γ) = g (calculated in 1.2.15) and dim M2(Γ) = g + m − 1 where g is the genus of XΓ 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

dim M2k(Γ) = dim S2k(Γ) + m. and s X dim S2k(Γ) = (2k − 1)(g − 1) + [k(1 − 1/ei)] + (k − 1) i=1

where x1, . . . , xs denote the elliptic points, e1, . . . , es their elliptic index and m is the number of inequivalent cusps. In absence of elliptic points, we have

dim S2k(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 S2k(Γ(1)) = [k/6] − 1 if k > 1 and k ≡ 1 mod 6 [k/6] otherwise and ( 0 if k = 2 dim M2k(Γ(1)) = dim S2k(Γ(1)) + 1 otherwise

In particular dim M4(Γ(1)) = dim M6(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 X (1.3.4) E (z) = (mz + n)−2k 2k 2 (m,n)∈Z2−(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 Z2 − (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 E2k is ζ(2k). We will choose a normalization so that the free coefficient be one −1 G2k(z) = ζ(2k) E2k.

The space M4(Γ(1)) is generated by G4, M6(Γ(1)) is generated by G6, M8(Γ(1)) is gener- 2 ated by G4, M10(Γ(1)) is generated by G4G6. In weight 12 there is the ﬁrst cusp form 3 2 ∆ = (G4 − G6)/1728. 3 Proposition 1.3.8. The rational function j : G4/∆ deﬁnes an isomorphism from XΓ(1) onto P1 C

Proof. By the discussion that precedes 1.3.5, there is a line bundle L on XΓ(1) such that 0 0 1 M12(Γ(1)) = H (XΓ(1), L). We know dim H (XΓ(1), L) = 2. Since XΓ(1) = P , L = OP1 (1). 3 It follows that G4 and ∆ as global section of L vanish exactly at one point. Morever as they are not proportional, their quotient define a morphism 3 1 j : G4/∆ : XΓ(1) → PC that is an isomorphism. 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 Ek has the form ∞ (2πi)k X (1.4.1) E (z) = ζ(z) + σ (n)qn. k (k − 1)! k−1 n=1 P k−1 where σk−1(n) = d|n d . 12 Proof. See [2, p.28] The terms with m = 0 in LHS sum up to the term ζ(z) in RHS. We have 1 X X 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 X (2πi)k X (1.4.2) (n − z)−k = nk−1e2πinz (k − 1)! n∈Z n∈N for all z ∈ H. 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 =(z) > 0 and it is L1 if k ≥ 2. Its Fourier transform is given by the formula Z ∞ fˆ(y) = (x − z)−ke2iπxydx. −∞ We can evaluate this integral by applying the residue formula to the 1-form (x−z)−ke2iπxydx. We get ( 2πi res ((x − z)−ke2πixydx) if y > 0 fˆ(y) = x=z 0 if y ≤ 0. The calculation of the residue gives ( k (2πi) yk−1e2πiyz if y > 0 fˆ(y) = (k−1)! 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 X (2πi)k X (1.4.3) (mz + n)−k = nk−1e2πimnz. (k − 1)! n∈Z n∈N If we sum the above formula over the positive integers m, we will get (1.4.1). P∞ n Corollary 1.4.3. Let Ek(z) = n=0 anq be the Fourier expansion of the Eisenstein series k−1 k−1 at ∞. There exists positive constants A, B > 0 such that An ≤ an ≤ Bn for every n ∈ N.

Proof. By the formula (1.4.1), it is enough to seek such an estimation for σk−1(n). In one side k−1 we have the obvious inequality n ≤ σk−1(n). On the other hand, we have the inequality

σk−1(n) X 1 = ≤ ζ(k − 1) nk−1 dk−1 d|n valid for k > 2. 13 P∞ n Proposition 1.4.4. If f = n=1 anq is a cusp form of weight k, its Fourier coefficients k/2 satisfy the inequality an ≤ Cn for some constant C independent of n. Proof. The equations (1.1.2) and (1.3.1) imply that the continuous function |f(z)yk/2| on H is Γ-invariant. It so defines a continuous function on the quotient Γ\H. Assume now −2πy Γ = SL2(Z). Since |q| = e , the vanishing of the constant terms of the Fourier expansion k/2 of f in the variable q implies that limy→∞ |f(z)y | = 0. It follows that the function k/2 |f(z)y | is bounded by a constant C1. For every natural integer n ∈ N, and for every y > 0, we have Z 1 −2πny −πinx −k/2 |an|e = | f(x + iy)e dx| ≤ C1y . 0 Let us pick y = 1/n and derive the inequality k/2 |an| < Cn 2π with C = e C1. This bound can be improved according to the Ramanujan-Peterson conjecture. P∞ n Theorem 1.4.5. Let f = n=1 anq is a cusp form of weight k of level N. Then for k−1 (n, N) = 1, we have an = O(n 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 S12(Γ(1)) of cusp forms of weight 12 for Γ(1) is one dimensional, its generator P∞ n is automatically an eigenvector. The ∆ function of Ramanujan ∆(z) = n=1 τ(n)q is a normalized cusp form whose Fourier coefficients are integers. Deligne proved the inequality |τ(p)| ≤ 2p11/2 that is the original conjecture of Ramanujan.

1.5. L-function attached to modular forms. If f ∈ Mk(Γ) is a modular form with P∞ n Fourier expansion f = n=1 anq . We call the Dirichlet series ∞ X −s (1.5.1) L(s, f) = ann n=1 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 (1.5.2) Λ(s, f) = (2π)−sΓ(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) = ikΛ(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) = iky−kf(i/y) It follows that f(iy) → 0 very rapidly as y → 0 too. It follows that the integral Z ∞ dy (1.5.5) f(iy)ys 0 y is convergent for all s and defines an analytic function of s. The following Mellin integral is absolutely convergent for <(s) > ν + 1 ∞ Z ∞ dy Z ∞ X dy (1.5.6) f(iy)ys = a e−2nπyys y n y 0 0 1 ∞ X Z ∞ dy (1.5.7) = a (2nπ)−s e−yys n y 1 0 ∞ −s X −s (1.5.8) = (2π) Γ(s) ann 1 = Λ(s, f)(1.5.9) P∞ −2nπy The exchange of the integration and infinite series is licit because the series 1 ane is P∞ −s absolutely convergent as well as 1 ann . 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).

ν Proposition 1.5.2. Let a1, a2 ... be a sequence of complex numbers which is O(n ) for some positive real number ν. Let L(s, f) be defined by the series (1.5.1), convergent for <(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. P∞ n Then f(z) = n=1 anq is a cusp form of weight k for Γ(1). ν P∞ n Proof. The assumption an = O(n ) implies that the series n=1 anq is absolutely for |q| < 1 or over the half-plane z = H if q = e2πiz. The Mellin integral ∞ Z ∞ dy Z ∞ X dy (1.5.10) f(iy)ys = a e−2nπyys y n y 0 0 1 ∞ X Z ∞ dy (1.5.11) = a (2nπ)−s e−yys n y 1 0 = Λ(s, f)(1.5.12) 15 is absolutely convergent for <(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 Z ∞ (1.5.13) f(iy) = Λ(σ + it, f)y−σ−itdt. 2πi t=−∞ 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−sss−1/2 as |s| → ∞ and <(s) ≥ δ > 0. On the vertical line s = σ + it for fixed σ > 0, we have √ |Γ(σ + it)| ∼ 2π|t|σ−1/2e−π|t|/2 as |t| → ∞. With the functional equation (1.5.3), we can transform (1.5.13) as follows 1 Z ∞ (1.5.15) f(iy) = ik Λ(k − σ − it, f)y−σ−itdt 2πi t=−∞ ∞ 1 Z 0 (1.5.16) = iky−k Λ((k − σ) + it0, f)yk−σ+it dt0 2πi t0=−∞ after the change of variable t0 = −t. We note that the line <(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 σ1 ≤ <(s) ≤ σ2 and =(s) > c. α Suppose that f(σ + it) = O(et ) for some real number α > 0. Suppose that for some M ∈ R, M M f(σ + it) = O(t ) for σ = σ1 or σ = σ2. Then f(σ + it) = O(t ) uniformly in σ ∈ [σ1, σ2]. Proof. See [5, p.118] and [10, p.124]. We can now apply the Cauchy theorem and move back the integration line to <(s) = σ 1 Z ∞ (1.5.17) f(iy) = iky−k Λ(σ + it, f)yσ+itdt 2πi t=−∞ and apply again the inverse Mellin transform (1.5.18) f(iy) = iky−kf(iy−1). 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) ⊂ Γ1(N) ⊂ Γ0(N) with (1.5.19) Γ(N) = {γ ∈ SL(2, Z) | γ ≡ 1 mod N} 1 ∗ (1.5.20) Γ (N) = {γ ∈ SL(2, ) | γ ≡ mod N} 1 Z 0 1 ∗ ∗ (1.5.21) Γ (N) = {γ ∈ SL(2, ) | γ ≡ mod N} 0 Z 0 ∗ For every N, there is an exact sequence × 1 → Γ1(N) → Γ0(N) → (Z/NZ) → 1. × It follows that the group (Z/NZ) acts on the space of modular forms Mk(Γ1(N)) as well × × as the space of cusp forms Sk(Γ1(N)). For every primitive character χ :(Z/NZ) → C , we will denote × (1.5.22) Mk(N, χ) = {f ∈ Mk(Γ1,N) | cf = χ(c)f ∀c ∈ (Z/NZ) }.

Similar notation Sk(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 0 −1 (1.5.23) S = N N 0

normalizes Γ0(N). Furthermore, it transforms the space Sk(N, χ) into Sk(N, χ¯) where χ¯ is the opposite Dirichlet character of χ. P∞ n P∞ n 2πiz Theorem 1.5.5 (Hecke). Let f(z) = n=0 anq and g(z) = n=0 bnq where q = e ν and an, bn are O(n ) for some real number ν. For positive integers k and N, the following conditions are equivalent k k/2 −k (A) g(z) = (−i) N z f√(−1/Nz). √ −s −s (B) Both ΛN (s, f) = (2π/ N) Γ(s)L(s, f) and ΛN (s, g) = (2π/ N) 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 a b Λ (s, f) + 0 + 0 N s 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 mod- ν ulo N. Suppose an, bn are sequences of complex numbers such that an, bn = O(n ) 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 La(s, µ) = P∞ −s P∞ −s −s n=0 µ(n)ann and Lb(s, µ¯) = n=0 µ¯(n)bnn . Let Λa(s, µ) = (2π) Γ(s)La(s, µ) and −s Λb(s, µ¯) = (2π) Γ(s)Lb(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, Λa(s, µ) and Λb(s, µ¯) have analytic continuation to the whole s-plane, are bounded in every vertical strips, and satisfy the functional equation 2 k τ(µ) 2 −s+ k (1.5.25) Λ (s, µ) = i µ(N)χ(D) (D N) 2 Λ (k − s, µ¯) a D b P∞ n Then f(z) = n=0 anq is a modular form of level N with nebentypus χ i.e. f ∈ Mk(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 | γ)(z) = det(γ)k/2(cz + d)−kf k 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 GL2 (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 GL2 (Q) acts naturally on the space Mk(Γ) as well as Sk(Γ). The construction of the algebra of double cosets of a congruence subgroup Γ in Σ = + GL2 (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 G (1.6.2) ΓαΓ = Γαi 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 Γα1 and Γα2 are in −1 position α if α1α2 ∈ ΓαΓ. Lemma 1.6.1 can be reformulated in an yet another way :

Lemma 1.6.2. For each left coset Γα1 and each double coset ΓαΓ, there are only ﬁnitely many left cosets Γα2 so that Γα1 and Γα2 are in position ΓαΓ.

Let HΓ denote the free abelian group generated by a basis indexed by the set of double cosets of Γ in Σ. We simple write Tα ∈ HΓ for the element in this basis indexed by the doubl coset ΓαΓ. We define a multiplication on HΓ by the following rule X α (1.6.3) Tα1 Tα2 = cα1,α2 Tα α α where cα1,α2 is the number of left coset Γβ such that Γβ1 and Γβ are in position α1 and Γβ and Γβ2 are in position α2, here Γβ1 and Γβ2 are fixed cosets in position α. The finiteness α of the numbers cα1,α2 follows immediately from 1.6.2. 18 F We define the action of ΓαΓ = i Γαi on Mk(Γ) by the formula X (1.6.4) f |k Tα = f |k αi. i

This formula deﬁnes an action of the Hecke algebra HΓ on the space of modular forms Mk(Γ). This action preserves the subspace of cusp forms Sk(Γ). We consider the Hecke algebra HΓ(1) with respect to the full modular group Γ(1) = SL2(Z) + as subgroup of Σ = GL2(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 GL2(Q) contains a unique diagonal matrix of the form d1 0 (1.6.5) α(d1, d2) = 0 d2

+ −1 with d1, d2 ∈ Q such that d1d2 is an integer. Proof. This can be best explained in term of relative position between lattices. A lattice of Q2 is a free abelian group of rank two contained in Q2. The map α 7→ α−1(Z2) deﬁnes a + 2 bijection between the set of left coset Γ(1)α in Σ = GL2 (Q) into the set of lattices of Q . If L, L0 ∈ Q2 are two lattices, the theorem of elementary divisors assert that there exist a basis 0 {x1, x2} of L such that {d1x1, d2x2} is a basis of L where d1, d2 are well deﬁned positive −1 rational numbers such that d1d2 is an integer.

Proposition 1.6.4. The algebra HΓ(1) is commutative. Proof. The transposed matrix g 7→ g> being an anti-homomorphism of Σ, it induces an anti-homomorphism on HΓ(1). On the other hand, as it ﬁxed the diagonal matrices α(d1, d2), it induces identity on HΓ(1). This means that HΓ(1) is a commutative algebra.

The Hecke operators are self-adjoint with respect to the Peterson inner product on SK (Γ(1)). Recall that this inner product is defined by the integral Z k dxdy (1.6.6) hf, gi = f(z)¯g(z)y 2 . Γ(1)\H y Here the invariant dxdy/y2 on H and defines a measure on the quotient Γ(1)\H, the expression f(z)¯g(z)yk is also invariant under Γ(1) and defines a function on Γ\H. Since f, g are cusp forms, f(z)¯g(z)yk 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/y2.

Theorem 1.6.5. The action of HΓ(1) on Sk(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) ∈ HΓ(1) by the following 19 formula X (1.6.7) T (n) = Tα(d1,d2). d1,d2∈N,d2|d1 d1d2=N The corresponding union of double cosets G (1.6.8) T (n) = Γ(1)α(d1, d2)Γ(1)

d1,d2∈N,d2|d1 d1d2=N is the set of integral matrices of determinant n

(1.6.9) T (n) = {α ∈ Mat2(Z) | det(α) = n}. P∞ m Proposition 1.6.6. Let f = m=1 amq be a cusp form of weight k for Γ(1). Let f |k P∞ m T (n) = m=1 bmq be the Fourier development of T (n)f. We have X k −k+1 (1.6.10) bm = n 2 d a md . a ad=n,a|m In particular

− k +1 (1.6.11) b1 = n 2 an. Proof. We can make explicit the action of T (n) on modular forms by decomposing T (n) in left cosets of Γ(1) G a b (1.6.12) T (n) = Γ(1) . 0 d a,b,d∈N ad=n,0≤b

2πimb P d The term 0≤b

Corollary 1.6.7. The Fourier coeﬃcient an of a normalized eigenform f of weight k with k −1 respect to Γ(1) is the eigenvalue of the operator n 2 T (n). 20 P∞ n Theorem 1.6.8. Let f = n=1 anq be a cuspidal eigenform of weight k for Γ(1) normalized so that a1 = 1. Then we have − k +1 (1.6.17) T (n)f = n 2 anf. In particular, we have the relation

(1.6.18) amn = aman for all relatively prime integers m, n ∈ N. Moreover, the Dirichlet series L(s, f) admits a development in Euler product ∞ X −s Y −s k−1−2s −1 (1.6.19) L(s, f) = ann = (1 − app + p ) . n=1 p Proof. Let c(n) denote the eigenvalue of T (n) with respect to the eigenvector f. We have b1 = c(n)a1 = c(n) since a1 = 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 ∞ ∞ X −s Y X −rs (1.6.20) L(s, f) = ann = ( apr p ). n=1 p r=0 The formula ∞ X −rs −s k−1−2s −1 (1.6.21) apr p = (1 − app + p ) 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 adèlesand the interpretation of modular form as automorphic forms on an adèlicgroup, we will postpone discussion after the adèlesbeing 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]: Sk(M, χ) → Sk(N, χ)

that associate to a form f ∈ Sk(M, χ) the form z 7→ f(dz) that belongs to Sk(N, χ). Let denote Sk(N, χ)old the subspace generated by the image of [d] for all integers d, M so that dM|N. The orthogonal complement of Sk(N, χ)old is denoted Sk(N, χ)new.

Theorem 1.7.1. (1) The space of new forms Sk(N, χ)new admits a basis consisting of normalized eigenform for the operators Tp 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 0 (3) If f, f ∈ Sk(N, χ)new such that for every prime p not dividing N, there exists ap such 0 0 0 that Tpf = apf and Tpf = apf 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 Cc(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 Cc(G) → C Z φ 7→ φ(g)dg. G The Haar measure also provides Cc(G) with a structure of algebra under the convolution product Z (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 Z π(φ)v = φ(g)π(g)vdg G for every φ ∈ Cc(G) and v ∈ H. The above integral can be defined as follows. We first consider the continous linear form Z v0 7→ φ(g)hπ(g)v, v0idg. G By Riesz representation theorem there exists a unique vector π(φ)v ∈ H so that Z hπ(φ)v, v0i = φ(g)hπ(g)v, v0idg. G

This deﬁnes an action of Cc(G) on H.

Deﬁnition 2.1.3. A sequence of positive functions φn is said to be approximating the delta function of the identity of G in the following sense

(1) φn ∈ Cc(G) are supported in a certain compact K of G R (2) φn(g)dg = 1 for every n G R (3) for every neighborhood U of 1G we have limn→∞ G−U φn(g)dg = 0. 22 Lemma 2.1.4. Let π be a representation of G on a Hilbert space H. If φn is a sequence of function approximating the delta function of 1G, then we have

lim π(φn)v = v n→∞ for all v ∈ H.

Proof. Let v ∈ V . For every > 0, there exists a neighborhood U of 1G so that for every R g ∈ U, ||π(g)v−v|| < . For an integer n large enough, we have limn→∞ ||φn(g)v||dg < R G−U since the family |φn| is bounded and limn→∞ φn(g)dg = 0. We can also check that R G−U U ||π(φn)v − v|| < 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 2πinx χn(x) = e . Let H = L2(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 Z −2πinx vn = π(χ−n)v = e π(x)vdx. T

Let Hn denote the image pn : v 7→ vn = π(χ−n)v.

(1) For n 6= m, the subspaces Hn and Hm are orthorgonal. (2) The space H is the Hilbert direct sum of its subspace Hn. Proof. Only the last statement is non trivial. For the last statement, it suﬃces to prove that PN Pn=N LN for all v ∈ H, v = limN→∞ n=−N pn(v). Since v− n=−N pn(v) is orthogonal to n=−N Hn, 0 LN for every v ∈ n=−N Hn, we have n=N 0 X |v − v | ≥ |v − pn(v)|. n=−N 0 LN It follows that we only need to construct a sequence vN ∈ n=−N Hn so that 0 lim |v − vN | = 0. N→∞

This can be achieved by constructing a delta sequence of functions φN on T so that for all 2πinx N, φN is a linear combination of e with |n| ≤ N. Classical examples of this is Fejer’s kernel N X |n| (2.2.1) K (x) = (1 − )e2πinx N N + 1 n=−N 1 sin2(π(N + 1)x) (2.2.2) = N + 1 sin2(πx) 23 2 On the complement of neighborhood of 0 ∈ T, sin (πx) > α for some given positive number −1 α which implies that KN (x) < e /(N + 1) for all N. This implies that Fejer’s sequence approximate the delta function. LN A vector v of some finite direct sum n=−N Hn 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. hπ(g)v, π(g)v0i = hv, v0i for all g ∈ K and v, v0 ∈ H. 0 Proof. [2, 2.4.3] Let hv, v i1 denote the given inner product on H. For all v, the function k 7→ 0 hπ(k)v, π(k)v i1 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 Z 0 0 hv, v i = hπ(k)v, π(k)v i1dk K is obviously positively definite. The inequalities C−1|v| < |π(k)v| < C|v| imply that it 0 defines the same topology on H as hv, v i1. 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 LX having X as the vector X at the origin. For every g ∈ G, the left translation map lg : G → G induces an isomorphism dlg : g → TgG and we have LX,g = dlg(X). Left invariant vector ﬁelds are equivalent to left invariant derivations. We have the following formula for the bracket

L[X,Y ]f = (LX LY − LY LX )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 7→ exp(tX) in an integral curve for the left invariant vector field LX . 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 d (L f)(g) = f(g exp(tX))| . X dt 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 Let us consider examples of finite dimensional representation of Lie algebras. Let G = SL2(R) and g = sl2(R) its Lie algebra. Let V be the standard 2-dimensional vector space on which G acts. For every integer k, Symk−1(V ) is a vector space of dimension k equipped with an induced action of G. This representation is algebraic and its derivation is a representation of Lie algebras g → gl(Symk−1(V )). But we are mostly interested in infinite dimensional representations. We define the universal enveloping algebra U(g) as the quotient of the tensorial algebra N L Nn g = N g by the ideal generated by [x, y] − x ⊗ y + y ⊗ x. Every representation π : g → End(V ) can be extended in a unique way into a homomorphism of algebras U(g) → End(V ). The Killing form is the symmetric bilinear form on g defined by B(x, y) = Tr(ad(x)ad(y)). Since it is invariant under the adjoint action of G on g, after derivation we get B([z, x], y) + B(x, [z, y]) = 0.

Proposition 2.3.4. Assume that the Killing form is non degenerate. Let x1, . . . , xd be a basis P of g and y1, . . . , yd be the dual basis so that B(xi, yj) = δij. Then the element Ω = i xiyi does not depend on the choice of the basis. Moreover, it belongs to the center of U(g). Proof. The adjoint action of G on g induces an action of G on g. Its derivation is an action of g on U(g). We can check that this action is X(A) = XA − AX. Sice Ω is ﬁxed under the action of G, it follows that it commutes with g. Since g generates U(g), Ω belongs to the center of U(g).

Let us consider the case g = sl2(R). Its standard base 1 0 0 1 0 0 (2.3.1) H = ,R = ,L = + 0 −1 + 0 0 + 1 0

satisfies the commutation rule [H+,R+] = 2R+,[H+,L+] = −2L+ and [R+,L+] = H+. The dual basis with respect to the Killing form is H+, 2L+, 2R+. The Casimir element is 2 (2.3.2) Ω = H+ + 2R+L+ + 2L+R+ It belongs to the center of the universal enveloping algebras. In fact the Casimir element is a generator of the center of the universal algebra of sl2. Proposition 2.3.5. The Casimir element Ω acts on the irreducible n-dimensional representation of g by the scalar n2 − 1. 2.4. Smooth, analytic and K-finite vectors in a Hilbert representation. A representation of a Lie group G on a topological vector space H is a continuous linear action π : G × H → H. If H is a Hilbert space, we call π a Hilbert representation. Observe that we do not require that the action of G preserve the inner product of H. If it does, we say that π is a unitary representation. A vector v ∈ H is C1 if for every X ∈ g, the limit 1 π(X)v := lim (π(exp(tX))v − v) t→0 t 25 exists. Recursively, we can define Ck-vectors for all k ∈ N. A smooth or C∞-vector v is a vector that is Ck for all k. The space of smooth vectors is a representation of the Lie algebra g. Proposition 2.4.1. Let π be a Hilbert representation of a Lie group G in a Hilbert space H. Then the subspace of smooth vectors Hsm is dense in H.

Proof. For a continuous function with compact support φ ∈ Cc(G), we deﬁne Z π(φ)v = φ(g)π(g)v dg. G If φ is a smooth function, π(φ)v is a smooth vector. The proposition follows from the existence of a delta sequence of smooth compactly supported function. Proposition 2.4.2. Let (π, H) be an irreducible unitary representation of G and Hsm the dense subspace of smooth vectors. Then all elements Ω ∈ Z(U(g)) acts on Hsm as a scalar.

Proof. This is an elaborate version of the Schur lemma. See [11, 1.6.5]. We can assume that the restriction of H to K is unitary. Then we have a decomposition of H as Hilbert direct sum Mˆ H = H(`). `∈Z The algebraic direct sum L H(`) is the subspace of K-ﬁnite vectors. `∈Z Proposition 2.4.3. For every ` ∈ Z, the subspace Hsm ∩H(`) is dense in H(`). The subspace of smooth, K-ﬁnite vectors is dense in H.

Proof. [11, 3.3.5].

We will also consider the subspace Han of analytic vectors. A vector v ∈ H is analytic if the function g 7→ π(g)v is analytic. It is equivalent to the apparently weaker condition that for all w ∈ H, the function g 7→ hπ(g)v, wi is analytic [12, p.278]. An analytic vector is obviously a smooth vector. The space Han is a subspace of Hsm which is table under the action of g. Proposition 2.4.4. Smooth, K-finite vectors are analytic. Proof. In order to prove that a function is analytic, it is enough to prove that it is annihilated by an elliptic differential operator. Recall that the Casimir element 2 2 2 2 Ω = H+ + 2R+L+ + L+R+ = H + (R+ + L+) − (R+ − L+) acts as scalar on smooth vectors. If v is K-eigenvector then it is also an eigenvector of 2 (R+ − L+). It follows that it is an eigenvector of the elliptic operator Ω + 2(R+ − L+) . Lemma 2.4.5. Let G be a connected Lie group. If V is a g-invariant subspace of Han, then its closure V¯ is a G-invariant subspace of H. Proof. See [11, 1.6.6]. If V ⊥ denote the orthogonal complement of V , then V¯ = (V ⊥)⊥. Let X ∈ g, v ∈ V¯ and w ∈ V ⊥. There exists > 0 such that if |t| < then X tn hπ(exp(tX))v, wi = hπ(X)nv, wi n! n 26 and the series converges absolutely. Since π(X)nv ∈ V and w ∈ V • it follows that hπ(exp(tX))v, wi = 0 for |t| < . The real analiticity of the function t 7→ hπ(exp(tX))v, wi implies that hπ(exp(X))v, wi = 0 for all v ∈ V¯ , w ∈ W and X ∈ g. It follows that V¯ is stable under exp(X) for all X ∈ g thus under G since exp(g) generates G. Let G be a real reductive Lie group, K a maximal compact subgroup of G anf g its Lie algebra. By a (g,K)-module, we mean a vector space V together with representation π of K and of g subject to the following conditions (1) V is a direct sum of finite dimensional irreducible representations of K (2) The actions of K and of g are compatible i.e. for every X in the Lie algebra of K and for every vector v ∈ V , we have the relation d (2.4.1) π(X)v = (exp(tX))f| . dt t=0 (3) For every g ∈ K and X ∈ g, we have the relation (2.4.2) π(g)π(X)π(g−1)v = π(Ad(g)X)v. The module is said to be admissible if each irreducible representations of K occurs with finite multiplicity. Proposition 2.4.6. Let (π, H) be an irreducible Hilbert representation of G. Then the subspace of smooth K-finite vectors V is a (g,K)-module. Proof. The commutation relation (2.4.1) and (2.4.2) can be checked upon the definition of π(X). If v is a K-finite vector, then so is π(X)v because the finite dimensional vector space generated by π(Y )π(k)v with Y ∈ g and k ∈ K is stable under the action of K. As in 2.2.2, the Hermitian form can be made K-invariant without changing the underlying topology of H. Proposition 2.4.7. Let V be a finitely generated (g,K)-module on which Ω acts as a scalar. Then V is admissible. Proposition 2.4.8. Let V be an irreducible (g,K)-module. Then the Casimir element Ω acts on it as a scalar. Proof. We first prove that V as a countable basis. For every v ∈ V , the vectors kv with k ∈ K span a finite dimensional subspace span(Kv). The space U(g)span(Kv) is then a nonzero (g,K)-submodule of V . Since V is irreducible, we have V = U(g)span(Kv) and in particular, V has a countable basis. The following version of Schur lemma, due to Dixmier, implies that Ω acts as a scalar. We will see later that from this fact, we can classify completely all irreducible (g,K)-modules and then derive their admissibility. Lemma 2.4.9. Suppose that V is countable dimensional and that S ⊂ End(V ) act irre- ducibly. If T ∈ End(V ) commutes with all elements of S then T is a scalar multiple of identity.

Proof. It is enough to prove that there exists x ∈ C so that T − c1V is not invertible. If it is the case, either its kernel or its image is a proper subspace which is stable under S. This would contradict the irreducibility. 27 If for all c ∈ C, T − c1v is irreducible. It follows that for all polynomial q(t) ∈ C[t], Q[T ] is invertible and therefore we can define the operator R[T ] for all rational fraction R(t) ∈ C(t). We derive a linear injective map C(t) → V defined by r(t) 7→ r[T ]v. This implies C(t) has a countable basis which is a contradiction. Proposition 2.4.10. Let (π, H) be a irreducible unitary representation of G. Then V = Hsm ∩ Hfin is an irreducible admissible (g,K)-module. Proof. Let v ∈ V a non zero vector. Then V 0 = U(g)span(Kv) is a (g,K)-module of finite type on which Ω acts as a scalar. It follows that its is admissible i.e for every ` ∈ Z, V 0(`) is finite dimensional. But we know that V 0 is a g-invariant subspace of Han, then V¯ 0 is G-stable thus V¯ 0 = H. It follows that H = Lˆ V 0(`). It follows that V (`) = V 0(`) is finite `∈Z dimensional and H is admissible. Two Hilbert representations (π, H) and (π0, H0) of G are said to be infinitesimally equivalent if their associated (g,K)-modules are isomorphic. A priori there exist non isomorphic irreducible Hilbert representations that are infinitesimally equivalent. But as we will see in the case of GL2(R), irreducible unitary representations that are infinitesimally equivalent, have to be isomorphic. + 2.5. (g,K)-modules for GL2(R). Let G = GL2(R) and K = SO2(R) its maximal compact group. cos θ sin θ K = | θ ∈ /2π . − sin θ cos θ R Z Since K is isomorphic to the circle group R/Z, all irreducible unitary representations of K are one dimensional and classified by the integers. For an integer ` ∈ Z, the `-th representation of K is given by the `-th power in the circle group. Let V be admissible (g,K)-modules. We have a decomposition of V into algebraic direct sum M V = V (`) `∈Z where K acts on V (`) by its `-th power. In sl2(C), we consider the following triple 0 1 1 1 i 1 1 −i (2.5.1) H = −i ,R = ,L = −1 0 2 i −1 2 −i −1 which is obtained from the triple (H+,R+,L+) by conjugation by the Cayley matrix i + 1 i 1 (2.5.2) c = − 2 i −1 −1 i.e. c (H+,R+,L+)c = (H,R,L). We also have eiθ 0 cos θ sin θ (2.5.3) c−1 c = . 0 e−iθ − sin θ cos θ Thus H is a generator of the complexified Lie algebra of K. In fact, the Lie algebra of circle group K is generated by the vector 0 1 (2.5.4) W = −1 0 28 and we have H = −iW . Since Ω is invariant under the adjoint action, we have (2.5.5) Ω = H2 + 2RL + 2LR. By Dixmier’s lemma 2.4.9, Ω acts on an irreducible (g,K)-module by a scalar. L Proposition 2.5.1. Let V be admissible (g,K)-modules and V = ` V (`) be is decomposition in direct sum of isotypical K-components. (1) V (`) is the space of vectors v ∈ V such that Hv = kv. (2) If v ∈ V (`), then Rv ∈ V (` + 2) and Lv ∈ V (` − 2). Moreover, if v 6= 0, then Rnv is a generator of V (` + 2n) and Lnv is a generator of V (` − 2n). (3) The dimension of V (`) is at most one. If V (`) 6= 0 and V (k) 6= 0 then k − l is an even integer. If V (`) 6= 0 for some even (resp. odd) integer, we say that V is an even (resp. odd) module. (4) Let Ω act on V by the scalar ω. For v ∈ V (`), we have (2.5.6) 4LRv = (ω − `(` + 2))v , 4RLv = (ω + `(2 − `))v. If v 6= 0 and Rv = 0 then ω = (`+1)2 −1. If v 6= 0 and Lv = 0 then ω = (`−1)2 −1. Proof. For v ∈ V (`), we have d W v = ei`tv| = i`v. dt t=0 It follows that Hv = kv. By using the commutation rule [H,R] = 2R, we have HRv = RHv + [H,R]v = (` + 2)Rv that implies Rv ∈ V (` + 2). We have similarly HLv = (` − 2)Lv. By using the commutation rule [R,L] = 2H, we have 4LRv = Ωv − H2v − 2[R,L]v = (ω − `(` + 2))v. The other affirmations follow from these computations. Definition 2.5.2. Let V be an irreducible admissible (g,K)-module. The K-type of V is the set Σ(V ) of integers k ∈ Z so that V (`) 6= 0. The central character of V is given by the eigenvalue α of Z. The infinitesimal character is given be the eigenvalue ω of Ω. Corollary 2.5.3. The isomorphism class of an irreducible admissible (g,K)-module is determined by its central character, its infinitesimal character and its K-type Σ(V ). Let ω be the eigenvalue of Ω on V (the infinitesimal character). Let us write ω = s2 − 1 with s ∈ C well determined up to a sign. Let ∈ {0, 1} be the parity of V : (1) If s in not an integer, or an integer congruent to modulo 2 then the K-type of V is Σ = {` ∈ Z | ` ≡ mod 2}. (2) If s is an integer not congruent to modulo 2, then the K-type of V can be either

Σ+ = {` ≥ |s| + 1 | ` ≡ mod 2}

Σ0 = {−|s| − 1 < l < |s| + 1 | ` ≡ mod 2}

Σ− = {l ≤ −|s| − 1 | ` ≡ mod 2}

Note that the case s = 0 is exceptional because in that case Σ0 = ∅. 29 In every case, there exists nonzero vector v` ∈ V (l) with l in the set of K-type a a good choice of s so that

Hv` = lv` s + 1 + ` Rv = v ` 2 `+2 s + 1 − ` Lv = v ` 2 `−2 In the first case, the choice of s between the two solutions of teh equation ω = s2 − 1 does not matter. In the second case, we have to choose positive s in the cases Σ± and negative s in the case Σ0. 2.6. Unitary (g,K)-modules. Let (π, H) be a unitary representation of G. The space of K-finite vectors V = Hfin is then a dense subspace of G which is equipped with a structure of (g,K)-module. Proposition 2.6.1. If V = Hfin is the (g,K)-module associated with a unitary representation then the action of K is unitary and the action of fg is anti-self-adjoint i.e. that for all X ∈ g, v, w ∈ V , we have hXv, wi = −hv, Xwi. Proof. We derive hXv, wi + hv, Xwi = 0 from the relation hexp(tX)v, exp(tX)wi = hv, wi by Leibnitz rule. 1 1 Recall that R = 2 (H+ + iX) and L = 2 (H+ − iX) where H+,X ∈ g are real matrices. In other words, R,L are complex conjugate matrices. It follows that hRv, wi = −hv, Lwi for all v, w ∈ V .

Proposition 2.6.2. If (π1, H1) and (π2, H2) are irreducible unitary representations of G, fin they are unitarily equivalent if and only if they are infinitesimally equivalent i.e. if V1 = H1 fin and V2 = H2 are isomorphic as (g,K)-modules.

Proof. Let k ∈ Z so that V1(`) 6= 0. Then V2(`) 6= 0. Choose non zero vector x1 ∈ V1(`) and x2 ∈ V2(`) so that hx1, x1i = hx2, x2i. It is enough to prove hLx1, Lx1i = hLx2, Lx2i and the n n same for R because L x1 and R x1 form a basis for V1. But this follows from the calculation

(2.6.1) hLx1, Lx1i = −hRLx1, x1i 1 (2.6.2) = (ω − `(` + 2))hx , x i 4 1 1 and the identical calculation for x2. Proposition 2.6.3. Let V be an infinite dimensional irreducible unitary (g,K)-module. Then the central element Z act on V by a purely imaginary scalar µ and the Casimir element Ω act by a real scalar ω. Moreover, if we write ω in the form ω = s2 − 1 then there are only four possibilities (1) s is purely imaginary (unitary principal series), 30 (2) s is real and −1 < s < 1 (complementary series), (3) s ∈ Z (discrete series and limit of discrete series), Proof. Since V is irreducible Z and Ω ought act by scalars. Since Z is skew-symmetric, 2 the scalar µ has to be a purely imaginary number. Since Ω = H+ + 2R+L+ + 2L+R+ is symmetric, the scalar ω has to be a real number. Assume that ω is not of the form s2 − 1 for an integer s, then ω < 0 (resp. ω < −1) if V is an even (resp. odd) module. In fact, if ω is not of the form s2 − 1 with integer s then V (`) 6= 0 for all even (resp. odd) integer k depending on the parity of V . Moreover, if v ∈ V (`) is a non zero vector, then Rv 6= 0. It follows from (ω − `(` + 2))hv, vi = hv, 4LRvi = −4hRv, Rvi < 0 that ω − `(` + 2) < 0 for all even (resp. odd) integer k if V is even (resp. odd). In the first case, we have ω < 0 by taking ` = 0. In the second case, we have ω < −1 by taking ` = −1. • If ω < −1, then s2 is a real negative integer i.e. s is purely imaginary. • If −1 ≤ ω < 0, then s is a real integer in the interval (−1, 1). In this case V must be an even module. • The remaining case ω = s2 − 1 with s ∈ N. If s ≥ 1, V is called a discrete series. If s = 0 then V is called limit of discrete series. In the subsequent paragraphs, we will construct unitary representations of G who (g,K)- modules fit with the three cases enumerated above. In order to complete the picture, we will have to construct unitary representations in three classes enumerated in Proposition 2.6.3 : unitary principal series, complementary series and discrete series. For later references, we we will name

•Ps the (g,K)-module in the unitary principal series, associated to a purely imaginary number ±s, •Cs the (g,K)-module in the complementary principal series, associated to a purely real number −1 < ±s < 1, •D±k the (g,K)-module in the discrete series. A (g,K)-module with a lowest weight k ∈ N is denoted Dk, in this case s = ±(k −1). A (g,K)-module with a lowest weight −k ∈ −N is denoted D−k, in this case s = ±(k − 1). + 2.7. Unitary principal series of GL2(R) . Let B denote the subgroup of triangular matrices of GL2(R) y x y 0 1 y−1x 1 y−1x y 0 (2.7.1) b = 1 = 1 1 = 2 1 0 y2 0 y2 0 1 0 1 0 y2 × with y1, y2 ∈ R , x ∈ R. On B, the left and right invariant measures can be expressed as follows dy1dy2dx dy1dy2dx dl(B)b = 2 and dr(B)b = 2 . |y1| |y2| |y1||y2| As δB(b)dl(B)b = dr(B)b, the modulus character δ is given by the expression −1 δB(b) = |y1||y2| . × × For all s1, s2 ∈ C and : {±1} → C , we have a character χ : B → C s1 s2 χ(b) = (sgn(y1))|y1| |y2| 31 which is unitary if and only if both s1, s2 are purely imaginary. For every character χ of B, let H0(χ) be the space of continuous functions f : G → C so that 1/2 f(bg) = δB(b) χ(b)f(g)

We have a representation of G on the Banach space H0(χ) equipped with the ∞-norm.

Lemma 2.7.1. Let H0() denote the space of continuous functions f : K → C so that s1 s2 f(−k) = (−1)f(k). If χ(b) = (sgn(y1))|y1| |y2| as above, the restriction to K defines a L linear bijection H0(χ) → H0. The subspace of K-finite vectors of H(χ) is Hn if is L n even trivial and n odd Hn if is non trivial. Proof. This follows from the Iwasawa decomposition G = BK and B ∩ K = {±1}. Proposition 2.7.2. Let H() be the space of L2-functions on K satisfying f(−k) = (−1)f(k). s1 s2 For every character χ(b) = (sgn(y1))|y1| |y2| , let H(χ) be the space of functions on G sat- 1/2 isfying f(bg) = δB(b) χ(b)f(g) so that the restriction to K is square integrable. Then the restriction to K defines an isomorphism H(χ) → H(). Moreover the right action of G on G H(χ) defines Hilbert representation IndB(χ) on the Hilbert space H().

The inner product for every f1, f2 ∈ H(χ) is given by Z ¯ (2.7.2) f1f2(k)dk K G G so that the restriction of IndB(χ) to K is unitary. But the representation IndB(χ) of G is not in general unitary. G Proposition 2.7.3. IndB(χ) is unitary if χ is unitary. ¯ ¯ Proof. For every f1, f2 the function f1f2(g) = f1(g)f2(g) satisﬁes the equation f1f2(bg) = ¯ δB(b)f1f2(g). As in C.2.1, there exists a canonical linear map

Cc(B\G, δB) → C that is G invariant. Moreover, this linear form is proportional to the integration over K by G C.2.2 so that the integral is G-invariant i.e. IndB(χ) is unitary. + By Iwasawa decomposition G = BK, any matrix g ∈ GL2(R) can be written uniquely under the form a b uy−1/2 0 y x cos θ sin θ (2.7.3) = c d 0 uy−1/2 0 1 − sin θ cos θ with z, y ∈ R+, x ∈ R and θ ∈ R/2πZ. Note that we have written the central matrix in a complicated way to make sure that det(g) = u2.

Proposition 2.7.4. For every χ = (, s1, s2), the space of K-finite vectors V (χ) in H(χ) is generated by the vectors f` −1/2 uy 0 y x cos θ sin θ s +s s+1 il`θ (2.7.4) f = u 1 2 y 2 e ` 0 uy−1/2 0 1 − sin θ cos θ indexed by even (resp. odd) integers ` if is trivial (resp. non trivial). Here we have introduced the notation s = s1 − s2. 32 The action of g on Hfin is given by the formulas s + 1 + ` s + 1 − ` (2.7.5) L f = lf , L f = f , L f = f H ` ` R ` 2 `+2 L ` 2 `−2 and LZ f` = (s1 + s2)f`. In particular, the Casimir element (2.5.5) acts as follows 2 Ωf` = (s − 1)f`. The proposition from direct calculations of Lie derivatives. The following tabulations can be found in [2, p.155] and [4, p.116] with slightly different coordinates. We follow Lang’s calculations.

Proposition 2.7.5. We have the following formula for the Lie derivatives LH , LR, LL on C∞(G) ∂ (2.7.6) L = −i H ∂θ ∂ ∂ i ∂ (2.7.7) L = e2iθ iy + y − R ∂x ∂y 2 ∂θ ∂ ∂ i ∂ (2.7.8) L = e−2iθ −iy + y + L ∂x ∂y 2 ∂θ

Proof. To all X ∈ g is attached a left invariant vector ﬁeld whose associated ﬂow is gX (t) = g exp(tX) on G. We have

φ(gX (t)) = φ(uX (t), xX (t), yX (t), θX (t)) and du (t) ∂φ dx (t) ∂φ dy (t) ∂φ dθ (t) ∂φ (2.7.9) L φ(g) = X | + X | + X | + X | . X dt t=0 ∂u dt t=0 ∂x dt t=0 ∂y dt t=0 ∂θ

dzX (t) dxX (t) dyX (t) dθX (t) We are thus about to calculate the derivatives dt |t=0, dt |t=0, dt |t=0 and dt |t=0 for any X ∈ g. We need to remember the formula relating the coordinates (z, x, y, θ) with the matrix entries a, b, c, d in the formula (2.7.3). Apply g to i ∈ H we get the equality ai + b = yi + x ci + d that allows us to express x, y as functions of a, b, c, d. We can also derive from the formula a b y cos θ − x sin θ y sin θ + x cos θ (2.7.10) = uy−1/2 c d − sin θ cos θ the equality d − ic = teiθ that allows us to calculate t as the modulus of d − ic and θ as its argument. In particular we have d − ic (2.7.11) = eiθ. |d − ic| Note that we also have the formula ad − bc (2.7.12) y = . |ci + d|2 33 0 1 For R = , we have + 0 0 1 t exp(tR ) = + 0 1 It follows that a at + b g (t) = R+ c ct + d ai + at + b g (t)i = R+ ci + ct + d

= xR+ (t) + iyR+ (t)

= zR+ (t) We go on to calculate the derivative of x, y with respect to t dz ad − bc R+ | = dt t=0 (ci + d)2 ad − bc (d − ci)2 = |ci + d|2 |d − ci|2 = ye2iθ after (2.7.11) and (2.7.12). In particular, we have the formulae 0 xR+ (0) = y cos 2θ 0 yR+ (0) = y sin 2θ We calculate the derivative of θ ct + d − ic iθ (t) = log R+ ((ct + d)2 + c2)1/2 (c2 + d2)1/2 c(c2 + d2)1/2 − cd(d − ic)(c2 + d2)−1/2 iθ0 (0) = R+ d − ic c2 + d2 c(d + ic) cd = − c2 + d2 c2 + d2 ic2 = c2 + d2 c2 θ0 (t) = R+ c2 + d2 = sin2 θ 2 The derivative of u vanishes because det(g) = u and det exp(tR+) = 1. From these calculations we obtain ∂ ∂ ∂ (2.7.13) L = y cos 2θ + y sin 2θ + sin2 θ . R+ ∂x ∂y ∂θ 0 1 For W = , we have −1 0 34 cos t sin t exp(tW ) = − sin t cos t uy−1/2 0 y x cos(θ + t) sin(θ + t) g exp(tW ) = 0 uy−1/2 0 1 − sin(θ + t) cos(θ + t)

and we get ∂ (2.7.14) L = . W ∂θ 1 0 For H = , we have + 0 −1

et 0 exp(tH ) = + 0 e−t aet be−t g exp(tH ) = + cet de−t iaet + be−t g exp(tH )i = + icet + de−t

= xH+ (t) + iyH+ (t) de−t − icet iθ (t) = log . H+ (c2e2t + d2e−2t)1/2 From this we obtain the Lie derivative ∂ ∂ ∂ (2.7.15) L = −2y sin 2θ + 2y cos 2θ + sin 2θ . H+ ∂x ∂y ∂θ

We can now calculate the Lie derivatives LH , LR, LL as linear combination with complex coeﬃcients of LR+ , LW , LH+ . We have

LH = −iLW ∂ = −i ∂θ i 1 L = iL − L + L R R+ 2 W 2 H+ ∂ ∂ i ∂ = e2iθ iy + y − ∂x ∂y 2 ∂θ i 1 L = −iL + L + L L R+ 2 W 2 H+ ∂ ∂ i ∂ = e−2iθ −iy + y + ∂x ∂y 2 ∂θ

according to (2.5.1). 35 + 2.8. Complementary series of GL2(R) . The construction of the unitary principal series was based on the fact that for purely imaginary numbers s1, s2, the induced representation H(s1, s2) affords a positively definite Hermitian inner form. In fact, for all copmlex numbers s1, s2 ∈ C there is a Hermitian pairing ∞ ∞ H (s1, s2) × H (−s¯1, −s¯2) → C. The construction of complementary series is based on the existence of an intertwining operator. ∞ ∞ H (s1, s2) → H (s1, s2) ∞ so that H (s1, s2) affords a Hermitian inner form if s1 = −s¯2 and s2 =s ¯1. The latter condition is equivalent with s1 + s2 ∈ iR and s = s1 − s2 ∈ R. We will construct the intertwining operator and prove that the induced inner form is positively definite is s = s1 − s2 ∈ (−1, 1). For every smooth vector f ∈ H(χ) with χ = (, s1, s2), we consider the integral Z ∞ 0 −1 1 x (2.8.1) (M(s)f)(g) = f g dx. −∞ 1 0 0 1

Proposition 2.8.1. Suppose that <(s) > 0. For every K-finite vector f ∈ V (, s1, s2), the integral (2.8.1) is convergent, and M(s)f is a K-finite vector in H(, s2, s1). If f is K-finite, then so is M(s)f, and f 7→ M(s)f defines a homomorphism of (g,K)-modules V (, s1, s2) → V (, s2, s1). Proof. Assume more generally that f is a smooth vector. Let us evaluate the integral M(s)f at g = 1. For that, we use Iwasawa decomposition 0 −1 1 x (1 + x2)−1 −x(1 + x2)−1 (2.8.2) = (1 + x2)1/2 k 1 0 0 1 0 1 θ(x) where k ∈ K. It follows that Z ∞ 2 − 1+s (2.8.3) M(s)f(1) = (1 + x ) 2 f(kθ(x))dx. −∞ If f is smooth, its restriction to K is bounded so that the above integral is absolutely convergent as long as <(s) > 0. For every g ∈ G, the integral M(s)f(g) is still absolutely convergent since one can replace f by its right translated by g. In fact, the operator M(s) commutes with the right translation by G as long as the integrals converge. In particular, M(s) commutes with the right action of K, and therefore it transforms a K-finite vector onto a K-finite vector. We derive from the commutation rule 0 −1 1 x y 0 y 0 0 −1 1 y−1y x (2.8.4) 1 = 2 1 2 1 0 0 1 0 y2 0 y1 1 0 0 1 that if f ∈ V (, s1, s2) then M(s)f ∈ V (, s2, s1). Proposition 2.8.2. Assume that <(s) > 0, then we have s s+1 `√ Γ( 2 )Γ( 2 ) (2.8.5) M(s)f`,s = (−i) π s+1+` s+1−` f`,−s. Γ( 2 )Γ( 2 ) 36 Proof. We already know that M(s)f`,s is a multiple of f`,−s so that it is enough to calculate the constant M(s)f`,s(1). See [2, p.230] for the calculation of the constant.

Theorem 2.8.3. The intertwining operator M(s): V (, s1, s2) → V (, s2, s1) for <(s) > 0 by the integral (2.8.1) can be meromorphically continued to the whole complex plane. It induces a positively deﬁnite inner form on H(, s1, s2) for all real numbers s = s1 − s2 satisfying −1 < s < 1. Proof. It follows from 2.8.2 that M(s) can be meromorphically continued to the whole complex plane. For every complex numbers s1, s2, there is a Hermitian pairing ∞ ∞ H (s1, s2) × H (−s¯1, −s¯2) → C deﬁned by Z ¯ (f1, f2) 7→ f1(k)f2(k)dk. K By C.2.2, this pairing is G-invariant. Now we have a meromorphic family of intertwining operators

M(s): V (s1, s2) → M(s2, s1).

If s1 + s2 is purely imaginary and s = s1 − s2 is real, then s1 = −s¯2 and s2 = −s¯1 so that ∞ we have a Hermitian inner product on H (, s1, s2). It remains to check that for all even integers ` the constants s s+1 `/2√ Γ( 2 )Γ( 2 ) (−1) π s+1+` s+1−` Γ( 2 )Γ( 2 ) are positive for all real numbers 0 < s < 1. Suppose ` ≥ 0. It suﬃces now to observe that s s+1 s+1+` s+1−` `/2 in this product Γ( 2 )Γ( 2 ) and Γ( 2 ) are positive, and the sign of Γ( 2 ) is (−1) . Without this explicit formula we can check that the inner form is positive deﬁnite as follows. If v is the generator of V (`) then we have hRv, Rvi = (`(` + 2) − ω)hv, vi Here since ` is even integer `(` + 2) ≥ 0. We also know that ω < 0. It follows that hRv, Rvi and hv, vi have the same sign.

+ 2.9. Discrete series of GL2(R) . We recall that on the upper half plan H, we have an 2 k dxdy invariant measure µ = dxdy/y . For every integer k ≥ 2, let µk = y y2 . We consider the vector space 2 (2.9.1) Hk = Lhol(H, µk)

of holomorphic functions on H which are square integrable with respect to the measure µk. The following proposition shows that Hk is complete with repect to the topology deﬁned by the L2-norm, and thus is a Hilbert space.

Lemma 2.9.1. Let {fn} be a sequence of holomorphic functions on the open disc D which 2 is L -convergent. Then fn converges uniformly to a holomorphic function on any compact set. 37 Proof. [4, p.182] Recall the Cauchy formula for a holomorphic function f the unit disc D 1 Z 2π f(reiθ) f(0) = iθ dθ 2π 0 re thus 1 Z 2π |f(reiθ)| |f(0)| ≤ dθ. 2π 0 r It follows that δ3 1 Z δ Z 2π |f(reiθ)| |f(0)| ≤ dθ. 3 2π 0 0 r It follows that |f(0)| and more generally the uniform norm is dominated by the local L1-norm and therefore the local L2-norm. Since the uniform norm is dominated by the L2-norm, a L2-convergent sequence fn is also for the uniform topology. The limit is necessarily a holomorphic function again by application of the Cauchy formula. Proposition 2.9.2. For every a b g = ∈ G c d let us deﬁne −1 k/2 −k (πk(g )f)(z) = f|kg = (ad − bc) (cz + d) f(gz).

Then πk is an irreducible unitary representation on Hd.

Proof. We have already seen that f 7→ f|kg deﬁnes a representation. We now verify that −1 2 0 0 0 0 πk(g ) preserves the L -norm. If z = gz with real coordinates z = x + iy , then we have ad − bc y0 = y |cz + d|2

0 according to (1.1.2) and µz = µz according to 1.1.2. Now, for every f ∈ Hk we have Z k −1 2 0 2 (ad − bc) k ||πk(g )f|| = |f(z )| 2k y µz H |cz + d| Z 0 2 0k = |f(z )| y µz0 H = ||f||2

which proves the unitarity of the representation πk.

With help of the Cayley transform, Hd is equipped with a convenient orthogonal basis. Recall that the Cayley map 1.1.3 z − i z 7→ w = z + i defines an analytic isomorphism C between the upper half plane H and the open unit disc D. Let us write w = u + iv in cartesian coordinates and w = reiθ in polar coordinates. 38 Proposition 2.9.3. The isomorphism maps the density µk of H on the density νk of H given by the formula 4(1 − |w|2)k−2dudv ν = . k |1 − w|2k In particular, the Cayley map induces an isomorphism of Hilbert spaces 2 2 L (H, µk) = L (D, νk). Proof. This is a direct consequence of (1.1.10) and (1.1.12). n k ∞ 2 Proposition 2.9.4. The functions {w (1−w) }n=0 form an orthonormal basis for L (D, νk). The functions z − in (2i)k φ = | n = 0, 1,... n z + i (z + i)k 2 form an orthonormal basis for L (H, µk). Proof. Since the functions wn(1 − w)k on D and (2i)k(z − i)n/(z + i)n+k correspond to n k ∞ each other via the Cayley transform, it is enough to prove that {w (1 − w) }n=0 form an 2 orthonormal basis for L (D, νk). n k We first verify that ψn = w (1 − w) are square integrable on D with respect to the iθ measure νk. In polar coordinate w = re , we have dudv = rdrdθ so that 4(1 − r2)k−2rdrdθ ν = . k |1 − w|2k For every n, the integral Z 2π Z 1 2 2n 2 k−2 ||ψn|| ≤ 4 r (1 − r ) drdθ 0 0 the latter integral being obviously convergent. We verify now that ψm and ψn are orthogonal with respect to the measure νk if m 6= n. The integral Z Z ¯ m n 2 k−2 ψm(w)ψn(w)dw = 4 w w¯ (1 − r ) drdθ D D Z 1 Z 2π = 4 rm+n(1 − r2)k−2 ei(m−n)θdq 0 0 vanishes if m 6= n. − Proposition 2.9.5. The infinitesimal class of πk is the (g,K)-module Dk of highest weight k. Proof. According to the classification of irreducible (g,K)-modules, it is enough to show that the K-type of πk is the set of integers {−k − 2n|n ∈ N}. It is enough to show that kθ ∈ K acts on the vector z − in (2i)k φ = n z + i (z + i)k −1 2iπ(k+2n)θ by πk(kθ )φn = e φn. This can be done by a direct calculation. 39 2 Let us do this calculation by writting down explicitly the action of G on L (D, νk). Con- jugate by the Cayley matrix, we get 1 1 −i a b 1 1 α β (2.9.2) = 2 1 i c d i −i β¯ α¯ where α = (a + bi − ci + d)/2 and β = (a − bi − ci − d)/2. We also have (ci − d)w + (ci + d) cz + d = 1 − w (−α + β¯)w + (¯α − β) = 1 − w a b If g = , then we have c d αw + β (|α|2 − |β|2)k/2(1 − w)k (2.9.3) π (g−1)ψ(w) = ψ k βw¯ +α ¯ ((−α + β¯)w + (¯α − β))k 2 for every ψ ∈ L (D, νk). If α β eiθ 0 = β¯ α¯ 0 e−iθ then we have eikθ(1 − w)k π (g−1)ψ(w) = ψ(e2iθw) . k (1 − e2iθw)k n k For ψn = w (1 − w) we have −1 i(2n+k)θ πn(g )ψn(w) = e ψn(w).

This implies that the K-type of πk is the set of integers {−k − 2n|n ∈ N}. We attach to each holomorphic function f on H the smooth function φ on G (2.9.4) φ(g) = f(gi)j(g, z)−k with the automorphy factor (2.9.5) j(g, z) = det(g)−1/2(cz + d). a b if g = . c d We will make the formula for φ more explicit with help of the Iwasawa decomposition. Every matrix g ∈ G can be written uniquely in the form uy−1/2 0 y x cos θ sin θ (2.9.6) g = 0 uy−1/2 0 1 − sin θ cos θ where x, y, u ∈ R with y, u > 0 and θ ∈ R/2πZ. Apply g to i ∈ H, we get gi = x + iy. The automorphy factor can be calculated to be j(g, i) = y−1/2e−iθ so that we get (2.9.7) φ(g) = yk/2eikθf(x + iy). The above construction f 7→ φ deﬁnes a bijection between smooth functions f on H and smooth functions φ on G that satisfy the equation ikθ (2.9.8) φ(gkθ) = e φ(g) 40 The opposite mapping φ 7→ f associates to a smooth function φ on G the function on H y x (2.9.9) f(z) = y−k/2φ y−1/2 0 1 for every z = x + iy with y > 0. Proposition 2.9.6. The above construction f 7→ φ deﬁnes a bijection between holomorphic functions f on H and smooth functions φ on G that satisfy ikθ (2.9.10) φ(gkθ) = e φ(g) and

(2.9.11) LLφ = 0 where LL is the left diﬀerential derivation attached to the nilpotent matrix L of (2.5.1).

Proof. We only to prove that f is holomorphic if and only if φ is annihilated by LL. If f is holomorphic, it is annihilated by the operator ∂ 1 ∂ ∂ (2.9.12) = + . ∂z¯ 2 ∂x ∂y According to 2.7.5, we have ∂ ∂ i ∂ L = e−2iθ −iy + i + L ∂x ∂y 2 ∂θ k/2 ikθ In order to prove that LL annihilates φ(g) = y e f(x + iy), it is enough to verify that it annihilates yk/2eikθ and f(x + iy). One can check ∂ ∂ i ∂ −iy + y + (yk/2eikθ) = 0 ∂x ∂y 2 ∂θ by direct calculation. The factor f(x+iy) is also annihilated because of the Cauchy-Riemann equation. Proposition 2.9.7. The map f 7→ φ deﬁned in (2.9.4) deﬁnes an intertwining operator 2 2 2 from L (H, µk) into L (G). In particular, L (H, µk) is a square integrable representation.

3. Automoprhic forms on SL2(R)

In this chapter, we denote G = SL2(R) and H the upper half plane. The main reference is [1]. 3.1. Siegel domains. Let Γ be a Fuschian group of first kind. Let us suppose that ∞ is a cusp for Γ. The group B of triangular matrices is the stabilizer of ∞ for the homographic 1 action of G on P (C). We have B = NA where N is the group of unipotent triangular matrices and A the group of diagonal matrices. The action of A on the Lie algebra of N × defines a character α : A → R . For each compact subset ΩN of N and t > 0, we define the Siegel domain S = ωAtK where At = {y ∈ A | α(y) > t}. If ω = {|x| ≤ h}, then the image of S is the upper half band {x + iy | |x| ≤ h and y > t}. 41 Assume that ω contains an interval in N = R which is a fundamental domain for the discrete subgroup Γ∞. Then Γ∞S is an upper half plane {x + iy | y > t}. We define by conjugation the notion of Siegel sets for other cusps.

Proposition 3.1.1. Let Γ be a Fuschian group of ﬁrst kind and let u1, . . . , um be a set of representatives of Γ-equivalence of cusps. There exists a compact subset C of H and a Siegel Sm domain Si for each cusp such that C ∪ i=1 Si contain a fundamental domain of Γ.

∗ Proof. For each cusp xi, we choose a Siegel domain Si whose image in Γ\H is a neighbor- ∗ hood of xi. Since Γ\H is compact by assumption, the complement of the union of images of Si is relatively compact. We can choose a compact C ⊂ H whose image contain it.

3.2. Growth condition. Assume that ∞ is a cusp for Γ and let Γ∞ is the stabilizer of ∞ in Γ. Let φ be a function on G that is Γ∞-invariant on the left and has K-type k for the action of K on the right i.e.

ikθ φ(γgkθ) = f(g)e for all γ ∈ Γ∞, kθ ∈ K. Then φ is said to be of moderate growth at ∞ if there exists λ > 0 so that

|φ(g)| < yλ

if g(i) = x + iy with x, y ∈ R and y > t for some ﬁxed t. In other words, this inequality is valid on a Siegel domain. The function φ is said to be of rapid decay if this inequality is valid for all λ.

Deﬁnition 3.2.1. A function φ :Γ\G → C is said to be of moderate growth if it is of moderate growth at every cusps of Γ.

∞ Lemma 3.2.2. If f has moderate growth of exponent λ then so does f ∗α for all α ∈ Cc (G). Lemma 3.2.3. Suppose that φ has moderate growth of exponent α on a Siegel domain S. ∞ α Assume that φ1 = φ ∗ α for some α ∈ Cc (G) then |Dφ(g)| < y for any D ∈ U(g) and g ∈ S. We will say that φ1 as above has uniform moderate growth. Proof. Since

Dφ1(g) = D(φ ∗ α)(g) = (φ ∗ Dα)(g). this lemma follows from the previous one. There is another way to deﬁne the condition of moderate growth. Let

||g||2 = a2 + b2 + c2 + d2.

Proposition 3.2.4. A function φ :Γ\G has moderate at the cusp of and only if there exists m > 0 such that |φ(g)| ≤ ||g||m. The function φ is of rapid decay if this inequality is valid for all m. 42 3.3. From modular forms to automorphic forms. Let f ∈ Mk(Γ) be a modular form of weight k with respect to Γ. The formula (2.9.4) defines a function φ on G that satisfies the equation and is annihilated by LL. Since the map f 7→ φ is G-equivariant, and f is Γ-invariant with respect to the action (1.6.1), φ is also invariant Γ-invariant. Thus φ defines a function Γ\G → C that satisfies (2.9.4) and is annihilated by LL. Proposition 3.3.1. Let f be a modular form of weight k and φ the smooth function on G defined by (2.9.4). Then φ satisfies the differential equation Ωφ = k(k − 2)φ.

Proof. It is clear for (2.7.16) that LH φ = kφ. In combining with LE− φ = 0, the calculation 2 (3.3.1) Ωφ = (H + 2E+E− + 2E−E+)φ 2 (3.3.2) = k φ + 2[E−,E+]φ (3.3.3) = k2φ − 2Hφ (3.3.4) = k(k − 2)φ. proves our proposition. Modular forms satisfy a condition of holomorphicity at the cusp, the associated function φ will satisfy a growth condition at the cusps that we are about to describe. Proposition 3.3.2. Let f be a modular form of weight k with respect to Γ. Then the function φ on G deﬁned as in (2.9.4) has moderate growth. If f is a cusp form, then φ has rapid decay. Proof. The function φ is given by the formula (2.9.7) phi(g) = yk/2eikθf(x + iy) where f admits the Fourier expansion ∞ X 2niπx −2nπy f(x + iy) = ane e . n=0

As y → ∞, φ(g) has moderate growth. If a0 = 0, φ(g) has rapid decay as y → ∞. Definition 3.3.3. A smooth function φ : G → C is an automorphic function for a Fuschian group of first kind Γ if (1) φ has a unitary central character (2) φ(γg) = φ(g) for all γ ∈ G and g ∈ G, (3) φ is K-finite on the right, (4) φ is Z(U(g))-finite, (5) φ has moderate growth at the cusps, Proposition 3.3.4. Automorphic form are real analytic. Proof. Because φ is both Z(U(g))-finite and K-finite, it satisfies an elliptic differential equation. It follows that φ is a real analytic function. Proposition 3.3.5 (Harish-Chandra). Let φ be an automorphic form. For every neighbor- ∞ hood U of identity there exists α ∈ Cc (G) with support contained in U so that φ = φ ∗ α. In particular, φ has uniform moderate growth. 43 Proof. Let V be the smallest closed G-invariant subspace containing φ. Since φ is K-finite and Z(U(g))-finite, V is admissible i.e. for all all integer `, the eigenspace space V (`) for the `-th power character of K is finite dimensional. ∞ Let Ic (G) denote the space of smooth function with compact support that are invariant ∞ with respect to the conjugation by K. We will prove that there exists a α ∈ Ic (G) with arbitrarily small support such that φ = φ ∗ α. P By assumption φ is a finite sum `∈L φ` with φ` ∈ V (`) with L a finite subset of Z. The ∞ convolution φ 7→ φ∗α preserves L and defines a continuous linear map Ic (U) → End(L). Its ∞ image is vector subspace is a finite-dimensional vector space hence closed. Let αn ∈ Ic (U) be a delta sequence whose image in End(L) tends to identity. It follows that identity can be ∞ represented in the form φ 7→ φ ∗ α for some α ∈ Ic (U). 3.4. L2-automorphic forms. The Peterson inner product (1.6.6) for cusp forms has a 0 natural transposition to the framework work of automorphic forms. Let f, f ∈ Sk(Γ) be cusp forms of weight k with respect to a Fuschian group of first kind Γ. Let φ and φ0 be functions on G associated with f and f 0 as (2.9.7). Then Z Z ¯0 k dxdy ¯0 f(z)f (z)y 2 = φ(g)φ (g)dg. Γ\H y ZΓ\G where dg is the Haar measure of G. Definition 3.4.1. Let ω be a unitary character of Z.A L2-automorphic form with central character ω is a function φ :Γ\G → C which transforms as ω under the action of the center so that |φ(z)| is a square integrable function on ZΓ\G.

Proposition 3.4.2. Suppose that ∞ is a cusp of Γ. Let α ∈ Cc(G). Then there exists a constant c so that

|(φ ∗ α)(g)| ≤ cy||φ||2 2 for all f ∈ L (Γ\G), all g ∈ NAtK and g(i) = x + iy. Proof. See [1, 5.7]. Assume that α is supported by C−1 where C is a compact subset of G. By deﬁnition Z (3.4.1) (φ ∗ α)(g) = φ(gx−1)α(x)dx G from which we derive the estimate Z (3.4.2) |(φ ∗ α)(g)| ≤ ||α||∞ |φ(h)|dh. gC

Let CN and CA be compact subsets of N and A such that KC ⊂ CN CAK. After writing g in Iwasawa’s form 1 x y 0 cos θ sin θ (3.4.3) g = y−1/2 0 1 0 1 − sin θ cos θ we have gC ⊂ gKC 1 x y 0 y 0 (3.4.4) y−1/2 (Ad C )( C )K. 0 1 0 1 N 0 1 A 44 + Assume that CN ∈ R and CA ∈ R are closed ﬁnite intervals. The image of this compact in H is a rectangle. The question is how many fundamental domains we need to cover this rectangle. Since y > t for some ﬁxed positive t the fundamental domain we need to ”cover in vertical direction” is bounded. The adjoint action of y on CN dilates in length with rate y. Therefore the number of fundamental domain we need to cover the rectangle is O(y). We get the estimate Z (3.4.5) |φ(h)|dh ≤ C1y||φ||1 gC

for some constant C1. Using Cauchy-Schwarz inequality and the finiteness of the volume of Γ\G, we get Z (3.4.6) |φ(h)|dh ≤ C2y||φ||2. gC THe proposition derives from this and (3.4.2). Corollary 3.4.3. If φ ∈ L2(Γ\G) satisfying the condition (1, 2, 3, 4) of 3.3.3, then it also satisfies the moderate growth condition (5). Proof. By 3.3.4 and 3.3.5, if φ satisfies (1, 2, 3, 4) of 3.3.3, then φ is real analytic end there ∞ exists φ ∈ Cc (G) with support arbitrarily close identity such that φ = φ ∗ α. We have the estimate |φ(g)| ≤ cy||φ||2

on the Siegel domain ΩN AtK of the cusp ∞. We have also similar estimates for other cusps. It remains now to apply 3.1.1. 3.5. Constant terms. Suppose that ∞ is a cusp of Γ. Let B = NA is the subgroup of triangular matrices in G, N its unipotent radical and ΓN = Γ ∩ N. Let φ be a ΓN -invariant function on G that is locally integrable. The constant term φB of f is, by deﬁnition the function Z (3.5.1) φB(g) = φ(ng)dn. ΓN \N

where the invariant measure is normalized so that the quotient ΓN \N has volume one. Since the constant term is deﬁned by integration on the left, this operation commutes with left-invariant diﬀerential operators as well as the convolution on the right

D(φB) = (Dφ)B and (φ ∗ α)B = φB ∗ α

for all D ∈ U(g) and α ∈ Cc(G). P∞ n Lemma 3.5.1. Let f be a modular form of weigh k and f = n=0 anq its Fourier expansion at ∞. Let φ denote the associate function φ(g) = ykeikθf(x + iy). Then the constant term k ikθ of φ is given by the formula φB(g) = y e a0. 1 1 Proposition 3.5.2. Let Lcusp(Γ\G) be the subspace of L (Γ\G) of integrable functions that 1 have vanishing constant terms at every cusp. Then Lcusp(Γ\G) is a closed subspace, stable under the convolution on the right of Cc(G). Proof. 45 Since Γ\G has ﬁnite volume, by applying the Cauchy-Schwarz inequality Z Z 1/2 Z 1/2 |φ(x)|dx ≤ |φ(x)|2dx dx Γ\G Γ\G Γ\G we have an inclusion L2(Γ\G) ⊂ L1(Γ\G), and the injection is continuous. We deﬁne 2 2 1 Lcusp(Γ\G) = L (Γ\G) ∩ Lcusp(Γ\G). It follows that

2 2 Proposition 3.5.3. Lcusp(Γ\G) is a closed subspace of L (Γ\G), stable under the convolution on the right of Cc(G).

1 Proposition 3.5.4. Let X1,X2,X3 be a basis of the Lie algebra of g. Let φ ∈ C (ΓN \G). Then there exists a constant c > 0, independent of f, such that 3 ! −1 X |(φ − φB)(g)| ≤ cy |LXi φ|B(g) i=1 where g(i) = x + iy. Proof. We have denoted 0 1 (3.5.2) R = + 0 0 which is a generator of the Lie algebra of N. We have Z 1 tR+ (3.5.3) (φB − φ)(g) = (φ(e g) − φ(g))dt. 0 But clearly, Z t tR+ uR+ (3.5.4) φ(e g) − φ(g) = (RR+ φ)(e g)du 0

where RR+ is the right invariant derivation attached to R+. We want to convert the above expression to left invariant derivation.

d d −1 (R φ)(euR+ g) = φ(euR+ etR+ g)| = φ(euR+ getAdg R+ )| R+ dt t=0 dt t=0 We will use again the Iwasawa decomposition 1 x y 0 cos θ sin θ g = y−1/2 0 1 0 1 − sin θ cos θ we have uR tAdg−1R uR ty−1Adk−1R e + ge + = e + ge θ + .

There are smooth functions c1, c2, c3 on K so that 3 −1 X kθ R+ = ci(kθ)Xi i=1 46 We have 3 uR+ −1 X uR+ (RR+ φ)(e g) = y ci(kθ)(LXi φ)(e g). i=1 The continuous functions on the compact set K are bounded by some constant c so that

3.6. Convolution on the cuspidal spectrum.

1 Proposition 3.6.1. Let α ∈ Cc (G). There exists a constant c(α) so that

|(φ ∗ α)(g)| ≤ c(α)||φ||2

2 for all φ ∈ Lcusp(Γ\G). Proof. Let P be a cuspidal subgroup of Γ and S s Siegel set relative to P . By 3.4.2, there exists a constant c1(α), depending only on α such that

(3.6.1) |(φ ∗ α)(g)| ≤ c1(α)y||φ||2

for g ∈ S and g(i) = x + iy. If X ∈ g and LX is the associated left invariant derivation, we have LX (φ ∗ α) = φ ∗ LX (α). Thus there exists a constant c(α, X) such that

(3.6.2) |(LX (φ ∗ α))(g)| ≤ c(α, X)y||φ||2. for g ∈ S. Since the translation on the left by the unipotent radical N of B keeps the coordinate y of g invariant, the integration over N gives

(3.6.3) |(LX (φ ∗ α))B(g)| ≤ c(α, X)y||φ||2.

We are now applying the inequality to the elements X1,X2,X3 of a basis of g. By assumption φB = 0 so that (φ ∗ α)B = 0, we have the estimate

(3.6.4) |(φ ∗ α)(g)| ≤ c(α)||φ||2

for some constant c(α) that depends only on α. 2 Proposition 3.6.2. For all α ∈ Cc (G), the convolution φ 7→ φ∗α defines a compact operator 2 on Lcusp(Γ\G). 47 Proof. Let φn be a sequence of function on Γ\G such that ||φn||2 ≤ 1. The estimate 3.6.1 implies that there exists a contant c(α) so that |(φn ∗ α)(g)| < c(α) for all g and all n. Thus the sequence φn ∗α is bounded with respect to the uniform norm. The same estimate applies to LX (φn ∗α) for all left invariant derivation LX . It implies that this family is equicontinuous. Now by application of the Arzela-Ascoli theorem and we can extract a subsequence φn0 ∗ α that converges locally uniformly to a function φ0 that is continuous and bounded. Since 0 2 Γ\G has finite volume, φ is square integrable. It remains only to recall that Lcusp(Γ\G) is 2 0 2 a closed subspace of L (G) so that φ ∈ Lcusp(Γ\G). The following general statement is due to Gelfand, Graev and PS. The following proof is due to Langlands. See [11]. Proposition 3.6.3. Let (π, H) be a unitary representation of G. If there exists a delta sequence φn on G such that π(φn) is compact operator on H then H is a Hilbert direct sum of irreducible unitary representations which appear with finite multiplicity. Proof. The idea is to use the spectral theory of compact self-adjoint operators. Recall that a compact self-adjoint operators have eigenvalues, the set of eigenvalues has no accumulation point except 0, and for each non zero eigenvalue, the corresponding eigenspace is finite dimensional. Let (π1, H1) be an irreducible unitary representation of G. Since φn is a delta sequence, π1(φn) 6= 0 for some n. There exists λ 6= 0 so that the λ-eigenspace of π1(φn) is nonzero. Since the λ-eigenspace of the compact self-adjoint operator π(φn) is finite dimensional, (π1, H1) appears in (π, H) with finite multiplicity. We will now prove that H has at least one irreducible subspace with the help of the compact self-adoint operators π(φn). For n large enough π(φn) 6= 0 has a nonzero eigenvalue λ. Let V denote the λ-eigenspace of π(φn) which is finite dimensional. Consider the non zero subspace V 0 ⊂ V so that V 0 = V ∩ H0 where H0 is a closed invariant subspace of H. Since V is finite dimensional, this family has minimal element for the inclusion order. Let V1 is a minimal element of this family. 0 0 Let H1 denote the intersection of all closed invariant subspaces H ⊂ H such that H ∩ V = V1. We will prove that H1 is irreducible. If it is not, we have a direct decomposition H1 = A ⊕ B where A, B are closed invariant subspaces. Since A and B are stable under π(φn), if we decompose a vector v ∈ V1 as v = a ⊕ b with a ∈ A and b ∈ B, then a, b are also λ-eigenvectors. It follows that

V1 = (V1 ∩ A) ⊕ (V1 ∩ B).

The minimality of V1 implies that either V1 ∩ A = V1 or V1 ∩ B = V1. But this contradicts with the minimality of H1. Let H0 be the closure of the sum of all irreducible subspaces in H and let H00 be the orthogonal complement of V . If H00 6= 0, it contains an irreducible subspace. This would 0 00 0 contradict with the contradict with the deﬁnition of H . Therefore H = 0 and H = H . 2 Theorem 3.6.4. The Hilbert space Lcusp(Γ\G) decomposes as a direct sum of irreducible unitary representations of G, each occurs with ﬁnite multiplicity. 3.7. Duality theorem.

48 Proposition 3.7.1 (Gelfand, Graev, PS). There is an isomorphism + 2 Sk(Γ) = HomG(Dk ,Lcusp(Γ\G)). + Proof. Let ﬁx a lowest weight vector vk of Dk . It is of K-weight k and annihilated by LL. + 2 2 Let α : Dk → Lcusp(Γ\G). Then α(vk) is an analytic L -function that is of K-weight K, annihilated by LL. By 2.9.6, φ comes from a cuspidal modular form f of weight k.

Let s be a purely imaginary number and let Ps be the unitary principal series. Let v0 be a vector of weight 0 in Ps. For any intertwining operator 2 α ∈ HomG(Ps,Lcusp(Γ\G)) the function α(v0) is a cuspidal analytic function on Γ\G that is right K-invariant. It satisfies the differential equation Ωφ = (s2 − 1)φ. It defines an analytic function f :Γ\H → C which is an eigenvalue of the Laplacian ∂2 ∂2 1 − s2 (3.7.1) ∆ = −y2 + then ∆f = f. ∂x2 ∂y2 4 Those are Maass modular forms, eigenvector for the Laplacians with eigenvalues λ ≥ 1/4.

Conjecture 3.7.2 (Selberg). For all s ∈ [−1, 1], and Cs the complementary series, we have 2 HomG(Cs,Lcusp(Γ\G)) = 0 for all congruence subgroup. This is actually false for some non congruence subgroup.

4. Smooth representations of GL2(Qp)

In this chapter, G = GL2(F ) where F is a local nonarchimedean field. Let OF denote the ring of integers of F . 4.1. Cartan and Iwasawa decompositions. As in the case of real Lie groups, Cartan and Iwasawa decomposition plays an important role in their representation theory. In the case of GL2, these decompositions can be described with linear algebras over the p-adic fields. Let B = AN denote the subgroup of upper triangular matrices, A is the group of diagonal matrices and N the group of unipotent upper triangular matrices. The group K(1) = GL2(OF ) is a maximal compact open subgroup of G. Every compact subgroup subgroup of G is conjugate to a subgroup of K(1) of finite index. Proposition 4.1.1 (Cartan decomposition). G pd1 0 G = K(1) d K(1). 0 p2 d1≤d2,di∈Z Proof. Let X denote the set of all lattices L ⊂ F 2. The group G acts transitively on X 2 and its stabilizer at the standard lattice L0 = OF is K. It follows that X = G/K(1). The Cartan decomposition now follows from the theory of elementary divisors : for every lattice d1 d2 L ∈ X, there exist a basis x1, x2 of L0 so that L = p x1OF ⊕ p x2OF where d1, d2 are integers satisfying d1 ≤ d2. 49 Proposition 4.1.2 (Iwasawa decomposition). G pd1 0 G = BK(1) = N d K(1). 0 p2 d1,d2∈Z

Proof. Let x1, x2 denote the standard basis of F . The group B upper triangular matrices is the stabilizer of the line generated by x1. For every lattice L ∈ X, we can define two integers d1 d1, d2 ∈ Z as follows : L1 = L ∩ x1F being a lattice of x1F , must be of the form p x1OF for some integer d1; the quotient L/L1 being naturally a lattice in x2F , must be of the form d2 p x2OF . It is not hard to check that there exists a triangular matrix n ∈ N so that pd1 0 L = n d L0. 0 p2 The Iwasawa decomposition follows. ∞ 4.2. Hilbert representations. Let Cc (G) denote the space of locally constant function ∞ with compact support. With a choice of Haar measure on G, Cc (G) is an algebra with respect to the convolution product. Be aware that this algebra is not equipped wit ha unit. ∞ We observe that every function f ∈ Cc (G) is biinvariant with respect to some compact open subgroup K. In other words, if HK denotes the algebra, with unit, of K-biinvariant functions with compact support, then ∞ [ Cc (G) = HK K the union being taken over all compact open subgroups of G. ∞ Let (π, H) be a Hilbert representation of G. The algebra Cc (G) acts on H by Z π(φ)v = φ(g)π(g)vdg. G K If φ ∈ HK , then π(φ)v ∈ H . A vector v ∈ H is said to be smooth if is stabilized by a compact open subgroup of G. Let V be the subspace of smooth vectors in H. The group G acts continuously on V equipped with the discrete topology. For all compact open subgroup K of G, the subspace V K of K-invariant vectors is a module of HK . Proposition 4.2.1. Let (π, H) be a Hilbert representation of G and let V denote its subspace of smooth vectors. Then H is irreducible if and only if V an irreducible smooth representation K of G, if and only if for every compact open subgroup K of G, V is an irreducible HK -module. Smooth representations of p-adic groups play a similar roles to (g,K)-modules of real Lie groups. 4.3. Unramified representations. Definition 4.3.1. An irreducible smooth representation of G is said to be unramified if V K(1) 6= 0 where K(1) is the maximal compact subgroup of G. Unramified representations are just irreducible H-modules. We have a complete description of the algebra H that allows us to describe its irreducible modules. By using Gelfand’s trick, one sees that H is commutative. In fact, we have a much more precise description of this algebra using Harish-Chandra’s constant terms. 50 Theorem 4.3.2 (Satake isomorphism). For every φ ∈ H, the constant term of φ along B is the function φB : A → C given by the formula Z 1/2 φB(a) = δB(a) φ(an)dn N the Haar measure dn on N is choosen so that K ∩ N has measure one. The constant term map defines an isomorphism from H onto the algebra of compactly supporte functions on A invariant under A ∩ K and under the action of the Weyl group W .

Proof. Orbital integral of a diagonal elements can be expressed as constant terms. Corollary 4.3.3. Let Aˆ denote the complex dual torus of A. There is an isomorphism of algebras between H and C[Aˆ]W . In particular, the unramified representations of G are in natural bijection with A/Wˆ . ˆ Corollary 4.3.4. Let G = GL2(C) denote the Langlands dual group of G = GL2(F ). The unramified representations of G are classified by the semisimple conjugacy classes of Gˆ. The trivial representation of G is certainly unramified. It corresponds to the homomorphism H → C given by Z φ 7→ φ(g)dg. G The associated conjugacy semisimple class in GL2(C) is the class of the matrix p 0 (4.3.1) 0 1 Proposition 4.3.5. If V is an unramified representation then its contragredient V 0 is also an unramified. R Proof. The linear form v 7→ K(1) π(k)v is K(1)-invariant. Proposition 4.3.6. Let V be an unramified representation of V and v ∈ V be a K(1)- invariant vector. 4.4. Smooth admissible representations. A representation of G on a complex vector space V is called smooth if every vector v ∈ V is stabilized by a compact open subgroup K. Lemma 4.4.1. Let V 0 be a subspace of V that stable under the action of K and irreducible as representation of K. Then V 0 is finite-dimensional. Proof. A vector v0 ∈ V 0 is stabilized by a finite index subgroup K0 of K. The subspace of V 0 generated by gv for a set of representatives K/K0 is K-stable. Since V 0 is irreducible, V 0 is generated by those vectors and hence finite-dimensional. Let K be a compact open subgroup of G. Let V be a smooth representation of G. For every finite dimensional irreducible representation (ρ, Vρ) of K be the sum of all K-invariant subspaces of V which are isomorphic to Vρ. The V can be decomposed as an algebraic direct sum M V = V (ρ). ρ∈Kˆ 51 V is said to admissible if and only if for every ρ ∈ Kˆ , V (ρ) is finite dimensional. A smooth representation of G is admissible if and only if dim(V K ) is finite for all compact open subgroups K of G. Theorem 4.4.2 (Harish-Chandra). All irreducible smooth representations of G are admissible. Let V be a smooth representation of G. The contragredient representation V 0 is the space of smooth vectors in the vector space of linear forms on V . L Proposition 4.4.3. Let V be a smooth admissible representation of G. Let V = ρ V (ρ) denote the decomposition of V into isotypical components with respect to a compact open subgroup K of G. Then the contragredient of G is an algebraic direct sum M V 0 = V (ρ)0 where V (ρ)0 = Hom(V (ρ), C). In particular, V 0 is admissible if V is. In that case, we have V = (V 0)0.

Theorem 4.4.4 (Gelfand-Kazhdan). The group G = GL2(Qp) is equipped with the outer automorphism g 7→ >g−1. Let (π, V ) be a irreducible admissible representation of G. Then the contragredient (π0,V 0) is isomorphic with (π00,V ) where π00(g) = π( >g−1). This theorem is surprisingly diﬃcult to prove. We will need in particular to introduce the ∞ notion of character of an admissible representation. For every φ ∈ Cc (G), there exists open compact subgroup G so that φ ∈ HK . The operator π(φ): V → V has image contained in the ﬁnite-dimensional space HK . The number

Trπ(φ) = Tr(π(φ)|V K ) does not depend on the choice of K. We thus deﬁne a linear map ∞ Trπ : Cc (G) → C hence a distribution on G.

Proposition 4.4.5. Two irreducible admissible representations (π1,V1) and (π2,V2) having the same character Trπ1 = Trπ2 , are isomorphic.

Proof. Let K be a compact open subgroup of G small enough so that V1 and V2 have nonzero K K K-invariant vectors. It follows that V1 and V2 are irreducible HK -modules having the same function trace. It follows that they are isomorphic as HK -modules (Bourbaki, see Lang’s discussion on the Jacobson density theorem). Hence V1 and V2 are isomorphic as smooth representations of G. In order to prove the Gelfand-Kazhdan theorem, we need to prove that (π0,V 0) and (π00,V ) ∞ 0 −1 00 > −1 have the same character. Let φ ∈ Cc (G). Let denote φ (g) = φ(g ) and φ (g) = φ( g ). By definition, π0(φ) and π(φ0) are adjoint operators so that they have the same trace. Also by definition, we have π00(φ) = π(φ00). It is thus enough to prove that 0 00 Trπ(φ ) = Trπ(φ ) or in other words that the distribution Trπ is transposition invariant. 52 Proposition 4.4.6. Over GL(2,F ), a conjugation invariant distribution is also transposition invariant. 4.5. Analysis on totally disconnected spaces. A td-space is a separated topological space in which every point has a basis of neighborhood given by compact open subsets. Lemma 4.5.1. Let X be a td-space. Let Y be a compact subset of X. From every covering of X by a open subsets of X, we can extract a finite family that, after refinement, forms a covering of Y by disjoint open subsets of X.

∞ We will denote S(X) = Cc (X) the space of complex valued locally constant functions with compact support on X. For every compact open subset U of X, we denote 1U the characteristic function of U. All function φ ∈ S(X) is a ﬁnite linear combination of such functions 1U . The vector space S(X) has a structure of algebra with respect to the pointwise multiplication. For every x ∈ X, let mx the ideal of S(X) of functions vanishing at x. Every element φ ∈ mx is a ﬁnite linear combination of 1U where U is a compact open not containing x. In fact, we have mx = S(X − x). We have an exact sequence of C-vector spaces

0 → mx → S(X) → C → 0. In fact, all morphism in this sequence are ring morphism. We have more generally the following proposition. Lemma 4.5.2. Let Y be a closed subset of X and U = X − Y . The we have an exact sequence 0 → S(U) → S(X) → S(Y ) → 0. Proof. Only the surjectivity S(X) → S(Y ) is not totally obvious. It follows in fact from 4.5.1. Definition 4.5.3. A S(X)-module M is called smooth if for every m ∈ M, there exists a compact open subset U of X such that 1U m = m. Let M be a smooth S(X)-module. We can define the fiber of M at a point x as follows

Mx = M/mxM

where mxM is the submodule of M generated by the elements φm with φ ∈ mx and m ∈ M.

Lemma 4.5.4. Let M be a smooth S(X)-module and x ∈ X. An element m ∈ mxM if and only if for all small enough compact open U containing x so that 1U m = 0.

0 Proof. If m = 1V m where V is a compact open subset not containing x, then for all compact 0 open U containing x but disjoint from V we have 1U m = 1U 1V m = 0. Let m ∈ M such that 1U m = 0 for all small enough compact open U containing x. By smoothness hypothesis there exists a compact open subset V such that 1V m = m. If x∈ / V , we are done. If x ∈ V , there exists a compact open U with x ∈ U ⊂ V such that 1U m = 0. Then 1V −U m = m.

Lemma 4.5.5. Let M be a smooth S(X)-module. Suppose that Mx = 0 for all x ∈ X. Then M = 0. 53 Proof. Let m ∈ M be an arbitrary element. There exists a compact open subset V so that 1V m = m. For every x ∈ V , since Mx = 0 there exists a compact open subset Ux containing 0 x and contained in V such that 1Ux x = 0. We can even require that 1Ux m = 0 for all compact 0 0 open Ux contained in Ux containing x. It follows that 1Ux = 0 for all compact open subset 0 0 0 0 Ux of Ux that may contain x or not because if x∈ / Ux, we can write 1Ux = 1Ux − 1Ux−Ux . By 0 applying 4.5.1, we can write 1V as finite sum of 1Ux which implies m = 0. Lemma 4.5.6. Let 0 → M 0 → M → M 00 → 0 be an exact sequence of smooth S(X)-modules. Then for every x ∈ X, we have an exact sequence 0 00 0 → Mx → Mx → Mx → 0. Proof. The right exactness is a general property of tensor product. Here, we also have the left exactness because Mx is not only the residual fiber but the fiber itself. More concretely, 0 0 let m ∈ M whose image m ∈ M satisfies mx = 0. There exists a compact open U containing 0 0 x such that 1U m = 0. This implies that 1U m = 0 and of course mx = 0. There are plenty of natural smooth S(X)-module. Proposition 4.5.7. Let Y → X be a continuous map of td-spaces. Then S(Y ) is naturally a smooth S(X)-module. Moreover, for every x ∈ X, we have S(Y )x = S(Yx). Proof. Let φ be a locally constant with compact support U in Y . The image of U in X is a compact subset of X that can be covered by a compact open subset U of X. Then we have 1U φ = φ. We have an exact sequence

0 → S(Y − Yx) → S(Y ) → S(Yx) → 0 where S(Y − Yx) = mxS(Y ). It follows that S(Yx) = S(Y )x. Proposition 4.5.8. Let Y → X be a continuous map between td-spaces. Let G be a td- group acting on Y , preserving the map Y → X and acting transitively on the fibers Yx. Let S(Y )G be the quotient of S(Y ) by the subspace S(Y )(G) generated by the functions of the form gφ − φ. Then S(Y )G is a smooth S(X)-module whose fibers S(Y )G,x is of dimension at most one. Proof. Since the action of G on S(Y ) commute with multiplication by S(X), the subspace S(Y )(G) is a S(X)-submodule. The module quotient is automatically smooth because S(Y ) is a smooth S(X)-module. The fiber S(Y )G,x is the quotient of S(Y ) by the subspace generated by mxS(Y ) and S(Y )(G). It is thus the quotient of S(Yx) by the image of S(Y )(G) in S(Yx) which is S(Yx)(G). Now, since G acts simply transitively on S(Yx), the space of G-invariant distribution on Yx is of dimension at most one. Proof. (of 4.4.6) Let Y denote G − Z the space of non central 2 × 2-matrices. Let Y → X = F × F × denote the characteristic polynomial g 7→ (Tr(g), det(g)). The group G acts transitively on the fibers of Y → X by conjugation. For every y ∈ Y , the centralizer Gy is commutative, in particular unimodular, hence the quotient G/Gy has a G-invariant measure. The S(X)-module S(Y )G has fibers of dimension one. The transposition defines an involution on this space that acts as identity on every fiber. It follows from 4.5.5 that it acts 54 trivially on the whole module S(Y )G. In other words, a conjugation invariant distribution on Y in transposition invariant. Since the similar statement is trivial for Z, the statement follows from the exact sequence 4.5.2. The same technique of proving 4.4.6 will permit us to prove an important property of Bessel distributions. Let us fix a non trivial additive charater ψ : F → C×. We call Bessel functions the locally constant functions b : G → C satisfying the transformation property

b(n1gn2) = ψ(n1)ψ(n2)b(g). We will be interested in the more general notion of Bessel distributions b : S(G) → C satisfying

b(l −1 rn φ) = ψ(n1)ψ(n2)b(φ). n1 2 Recall that by definition l −1 rn φ(g) = φ(n1gn2). n1 2 > Let θ : G → G be the anti-involution defined by θ(g) = w0 gw0 where w0 is the permu- tation matrix. For every n ∈ N we have θ(n) = n. The involution θ induces an involution on S(G) and on the space of distribution. Proposition 4.5.9. If b is a Bessel distribution, then we have θ(b) = b. 4.6. Uniqueness of the Whittaker model. Let N be the subgroup of unipotent upper triangular matrices. Let ψ : F → C× be a non trivial character of F . A Whittaker function with respect to ψ is a locally constant function φ : G → C such that φ(ng) = ψ(n)φ(g) for all n ∈ N and g ∈ G and there exists a compact open subset C ⊂ G so that the support of φ is contained in NC. The space of all Whittaker functions is G cIndN (ψ) the compact induction of ψ. This space affords with a representation of G by translation on the right. Let (π, V ) be an irreducible admissible representation of G. A Whittaker model of G is an G intertwining operator α : V → cIndN (ψ). For all such α, we have a linear form lα : V → C given by lα(v) = α(v)(1G) which satisfies la(π(n)v) = ψ(n)la(v). In general, la is not fixed by any compact open subgroup so that lα does not belong to the contragredient representation V 0. We have in fact a bijection G HomG(V, cIndN (ψ)) = HomN (V, ψ)

by application of the Frobenius reciprocity. Let Vψ(N) be the subspace of V spanned by the vectors of the form π(n)v − ψ(n)v with n ∈ N and v ∈ V . Let VN,ψ = V/Vψ(N). Every linear functional l ∈ HomN (V, ψ) must factorize though VN,ψ and in fact we have

HomN (V, ψ) = Hom(VN,ψ, C). If these vector spaces are nonzero then we say that V admits a Whittaker model. Lemma 4.6.1. If V admits a Whittaker model with for a non trivial additive character ψ : F → C×, then it has Whittaker model for any non trivial additive character. Proof. The group A of diagonal matrices acts transitively on the set of non trivial characters of N. 55 Proposition 4.6.2. Let V be an irreducible admissible representation. If V admits a model of Whittaker then so does its contragredient V 0. Proof. We know by Gelfand and Kazhdan that the contragredient representation (π0,V 0) is isomorphic to (p00,V ) where π00(g) = π( >g−1). If l : V → C is a Whittaker functional of π with respect to (N, ψ) then it is a Whittaker functional for π00 with respect to >(N, ψ)−1. 00 After conjugation, we will get a Whittaker functionalof π with respect to (N, ψ). Theorem 4.6.3. An irreducible admissible representation admits at most one Whittaker model. In other words G dimG(V, cIndN (ψ)) ≤ 1. The theorem follows from 4.6.2 and the following proposition. Proposition 4.6.4. Let V be an irreducible admissible representation. Then we have G 0 G dimG(V, cIndN (ψ)) dimG(V , cIndN (ψ)) ≤ 1. Proof. Let l0 : V → C and l : V 0 → C be nonzero Whittaker functionals of V and V 0. If l, l0 are smooth linear form i.e. l0 ∈ V 0 and l ∈ V we can define the matrix coefficient function G → C c(g) = hl, π0(g)l0i that satisfies

c(n1gn2) = ψ(n1)ψ(n2)c(g). But in fact l, l0 are not smooth vectors, we will have to define c as a Bessel distribution. For every φ ∈ S(G), we consider the linear form π0(φ)l0 : V → C defined by v 7→ l0(π(φ0)v) where φ0 is the function φ(g) = φ0(g−1). This is a smooth linear form on V so that π0(φ)l0 ∈ V 0. In the same way, for every φ ∈ S(G), we have a smooth vector π(φ)l ∈ V . One can check that 0 0 0 0 0 (4.6.1) π(rnφ)l = ψ(n)π(φ)l and π (rnφ)l = ψ(n )π (φ)l We also have 0 0 0 0 π(lgφ)l = π(g)π(φ)l and π (lgφ)l = π (g)π (φ)l so that the map φ 7→ π(φ)l and φ 7→ π0(φ)l0 are intertwining operators S(G) → V and S(G) → V 0 respectively. We have a bilinear form on S(G) 0 0 b(φ1, φ2) = hπ(φ1)l, π (φ2)l i which satisfies

b(lgφ1, lgφ2) = b(φ1, φ2). Note that S(G) ⊗ S(G) = S(G × G). Consider the diagonal action of G acting by left translation on G × G given by the formula g(g1, g2) = (gg1, gg2). The quotient G\(G × G) −1 can be identified with G by the map G × G → G given by (g1, g2) 7→ g1 g2. The integration along the fibers of this map defines a linear map S(G) ⊗ S(G) → S(G) 56 0 0 −1 which is (φ1, φ2) 7→ φ1∗φ2 where φ1(g) = φ1(g ). Because the linear form φ1⊗φ2 7→ b(φ1, φ2) is G-invariant with respect to the diagonal action of G by lest translation on G × G, there exists a unique linear form c : S(G) → C 0 such that c(φ1 ∗ φ2) = b(φ1, φ2). Because of (4.6.1), c is a Bessel distribution −1 c(ln1 rn2 φ) = ψ (n1)ψ(n2)c(φ). By applying 4.5.9, we know that θ(c) = c where θ is the induced operaotr on the space of > distribution of θ(g) = w0 gw0. Thus 0 0 0 c(φ1 ∗ φ2) = c(θ(φ1 ∗ φ2)) = c(θ(φ2) ∗ θ(φ1)) thus 0 0 b(φ1, φ2) = b(θ(φ2), θ(φ1)). 0 0 0 If φ2 is in the kernel of the intertwining operator l : S(G) → V the θ(φ2) is in the kernel of l : S(G) → V . It follows that l0 determines the kernel of l. By Schur lemma, as V is 0 irreducible, l determines l up to a scalar. The proposition follows.

Lemma 4.6.5. For every compact subgroup N0 ⊂ N, we set Z −1 Vψ(N0) = {v ∈ V | ψ (n)π(n)vdn = 0} N0 Then we have V (N) = S V (N ). ψ N0 ψ 0

Proof. An element v ∈ Vψ(N) is a linear combination of vectors of the form π(n0)v0−ψ(n0)v0. Every n0 ∈ N, there exists a compact subgroup N0 of N such that n0 ∈ N0. Then we have Z −1 (4.6.2) ψ (n)π(n)π(n0)v0 − ψ(n0)v0dn = 0 N0 thus π(n )v − ψ(n )v ∈ V (N ). It follows that V (N) ⊂ S V (N ). 0 0 0 0 ψ 0 ψ N0 ψ 0 Let v ∈ V (N0). There exists a compact open subgroup N1 ⊂ N0 that stabilizes v. By shrinking N1 we can also assume that ψ is trivial on N1. Then we have Z X (4.6.3) ψ−1(n)π(n)vdn = ψ−1(n)π(n)v = 0 N0 n∈N0/N1

Let r denote the cardinal N0/N1. Then we have 1 X v = ψ−1(n)(ψ(n)v − π(n)v) r n∈N0/N1 and thus v ∈ Vψ(N). Let V be an admissible representation of G. The above lemma suggest that Whittaker functionals are fibers of V with respect to a structure of S(F )-module that we are going to define. Choose a non trivial additive character ψ : F → C×. The Fourier transform Z φˆ(x) = φ(xy)ψ(x)dx F 57 defines an isomorphism S(F ) → S(F ). There exists a unique choice of Haar measure dx so ˆ that φˆ(x) = φ(−x). The Fourier transform turns pointwise multiplication into convolution product. It follows that if we let φ ∈ S(F ) act on V by Z (4.6.4) ι(φ)v := φˆ(n)π(n)vdn N then V has a structure of S(F )-module. Lemma 4.6.6. V is a smooth S(F )-module.

Proof. Since V is smooth as representation of G, there exists a compact open subgroup N0

that stabilizes v. The Fourier transform of 1N0 is of the form 1U for certain compact open

subgroup U. Then we have ι(1U )v = π(1N0 )v = v. × Lemma 4.6.7. For every a ∈ F , Va and VN,ψa , where ψa is the additive character x 7→ ψ(ax) are the same quotient of V .

Proof. The description of Vψa (N) given in 4.6.5 identiﬁes this submodule with V (a). It

follows an identiﬁcation with quotient modules Va = VN,ψa . Proposition 4.6.8. Let 0 → M 0 → M → M 00 → 0 be an exact sequences of admissible representations. Then we have an exact sequence

0 00 0 → MN,ψ → MN,ψ → MN,ψ → 0. Proposition 4.6.9. Let V be an irreducible admissible representation. Then G has Whit- taker model.

Proof. Assume that V does not have Whittaker model. Then Va = VN,ψa = 0. It follows that V = V0. The ﬁber V0 is the Jacquet module of V that is ﬁnite dimensional as we will later show.

4.7. Jacquet module. Let (π, V ) be a smooth representation of G. The space V (N) is the subspace of V spanned by the vectors of the form π(n)v − v with n ∈ N and v ∈ V . For a compact subgroup N0 ⊂ N define V (N0) to be Z V (N0) = {v ∈ V | π(n)vdn = 0}. N0 As in 4.6.5, we have V (N) to be V (N) = S V (N ). Let define the Jacquet module N0 0 VN = V/V (N) which is also the fiber V0 of V as S(G)-module defined in (4.6.4). The Jacquet module VN is equipped with an action πN of A = B/N. Observe that if a vector v ∈ V is fixed by a compact open subgroup K of G, its imagev ¯ in VN is necessarily fixed by K ∩ A. Hence, (πN ,VN ) is a smooth representation of A provided that (π, V ) to be a smooth representation of G.

Proposition 4.7.1. VN is an admissible representation of A. The proof of the theorem is based on the notion of Iwahori decomposition. This decomposition is valid for an important family of compact open subgroups of G. Let denote 58 a b 1 0 K(pr) = ∼= mod pn c d 0 1 a b 1 ∗ K (pr) = ∼= mod pn 1 c d 0 1 a b ∗ ∗ K (pr) = ∼= mod pn 0 c d 0 ∗ For r ≥ 1, all these groups satisfy the following properties Lemma 4.7.2. Let K be one of the above compact open subgroups for an integer r ≥ 1. − Then every k ∈ K can be written uniquely in the form r = nan where n ∈ NK = N ∩ K, − − − a ∈ AK = A ∩ K and n ∈ NK = N ∩ K is a unipotent lower triangular matrix. × Proof. The existence of the Iwahori decomposition in the above example is d ∈ OF in all cases. Proposition 4.7.3. Let K be a compact open subgroup that admit an Iwahori decomposition as above. Then the canonical map K AK V → VN is surjective.

AK Proof. Let E be a finite dimensional subspace of VN . Since AK is a compact subset, there exists a finite dimensional subspace E0 of V AK that maps onto E. Since E0 is finite − − − dimensinoal there exists a compact subgroup N0 of N that fixes N . There exists a ∈ A −1 − 0 such that π(a)N0π (a) ⊂ NK hence AK NK fixes π(a)E . In other words, for every v ∈ π(a)E0 we have Z Z π(a)π(n−)vdn−da = v. AK NK It follows that Z Z Z π(n)π(a)π(n−)vdn−dadn = π(n)vdn. NK NK NK 0 K K The application v 7→ π(1NK )v send π(a)E into V . It follows that the image of V in VN contains πN (a)E. But since πN (a) is invertible in VN , it follows that K dim(E) = dim(πN (a)E) ≤ dim(V ).

AK Since E have been chosen as an arbitrary finite dimensional subspace of VN , it follows that AK K AK K AK VN is finite and bounded by dim(V ). We can thus take E = VN hence V → VN is surjective. K AK In fact, the structure of the map V 7→ VN can be made very precise because it is linear over a rather large algebra. Let A− denote the sub semigroup of A of diagonal matrices −1 − (a1, a2) such that a1a2 ∈ OF in other words A = {a ∈ A | |α(α)| ≤ 1} where α is the positive root. For a ∈ A, the conjugation by A contracts N and dilates N −. More precisely, we have −1 −1 − − (4.7.1) aNK a ⊂ NK and a NK a ⊂ NK − − Let C[A /AK ] be the algebra generated by the monoid A /AK . 59 Lemma 4.7.4. Let K be as above. The application a 7→ 1KaK defines a ring morphism − C[A /AK ] → HK . Proof. Using (4.7.1), we have a1Ka2 ⊂ Ka1a2K − for all a1, a2 ∈ A . The lemma follows.

K AK − Proposition 4.7.5. The map V → VN is C[A /AK ]-linear. There is a decomposition K K AK V = E0 ⊕ E1 where π(a) = 0 on E0 and is bijective on E1. The map V → VN is null on AK E0 and induces a bijection E1 → VN .

I AI Corollary 4.7.6. The map V → VN is a bijection if I = K0(p) is the Iwahori subgroup.

Proof. In this case 1IaI is an invertible element of the Hecke-Iwahori algebra HI .

In our case G = GL2(F ), a much stronger statement is true.

Proposition 4.7.7. If V is irreducible then the Jacquet module VN is ﬁnite dimensional. In fact we have dim(VN ) ≤ 2.

Proof. We first prove that VN contains a one-dimensional representation of A. We already know that VN is a smooth admissible as representation of A. For some compact open AK subgroup AK of A, VN is non zero and finite dimensional. The commutative group of finite type A/AK acting on this finite dimensional vector space admits at least one eigenvector. × Thus there exists a non zero vector v ∈ VN and a character χ : A → C so that for every a ∈ A, πN (a)v = χ(a)v. Now we have a non zero B-map V → χ. We derive from the Frobenius reciprocity a G non zero map V → IndB(χ) which is necessarily injective since V is irreducible. Since the G Jacquet functor is exact, we have an injective map VN → IndB(χ)N . It is thus enough to prove the following statement. G Proposition 4.7.8. dim IndB(χ)N = 2 for all character χ of A. G Proof. It amounts to prove that the space of linear functional L : IndB(χ) → C such that G G L(πχ(n)v) = L(v) for all v ∈ IndB(χ), n ∈ N and πχ is the representation of G in IndB(χ). Let L be such a functional which is nonzero. The space of the induced representation consists of function f : G → C so that for all b ∈ B, we have f(bg) = χ(b)f(g). We have a linear map G Λ: S(G) → IndB(χ) defined by Z Λ(φ)(g) = φ(bg)δ−1(b)χ−1(b)db B where db is the left invariant measure on B. We are now considering the distribution φ 7→ L(Λ(φ)). It satisfies the property

(4.7.2) lbrn(L ◦ Λ) = δ(b)χ(b)(L ◦ Λ)

Recall the Bruhat decomposition G = Bw1B t B where Bw1B is an open subset and B is closed. In applying 4.5.2, it is enough to prove that on Bw1B and on B there is exactly on 60 distribution satisfying (4.7.2). This is easy to see because B × N, B acting on the left and N on the right, acts transitively on each Bruhat cell. One important consequence of the finiteness of the Jacquet module is the existence of Whittaker model for infinite dimensional representations. Corollary 4.7.9. Infinite dimensional smooth admissible representations of G have Whit- taker model. Proof. Let V be an infinite dimensional irreducible admissible representation of G. Assume that V has no Whittaker model for some non trivial additive character ψ. Using the transitive action of A on the set of non trivial characters of N, we conclude that V has no Whittaker model for any non trivial additive character of N. As S(F )-module 4.6.4, we have Vx = 0 for all x 6= 0. It follows that map V → V0 is an isomorphism. We know that the fiber of V over x = 0 is the Jacquet module VN which is finite dimensional. It follows that V is finite dimensional. 4.8. Principal series. A smooth character χ : F × → C× is a character that is trivial on × × × Z a compact open subgroup of F . Recall that F = OF × p . The character χ consists in 0 × × a finite order character χ : OF → C and a complex number s well determined modulo 2iπ log(p) such that χ(p) = p−s. The character χ is unitary if |z| = 1. The character is said to be unramified if χ0 = 1. × × × Let χ : A → C be a smooth character. Since A = F × F , χ = (χ1, χ2) where χ1, χ2 are characters of F × as above. We say χ is unramified if χ is trivial on A∩K which amounts to saying that both χ1 and χ2 are unramified. We define the principal series representation G iB(χ) as the induces representation G G 1/2 iB(χ) = IndB(χ ⊗ δ ). Recall that this is the space of locally constant functions f : G → C such that 1/2 (1) for every b ∈ B, f(bg) = δB (b)χ(b)f(g), (2) there exists a compact open subgroup K of G so that f(gk) = f(g) for all k ∈ K. G The translation of G on the right defines an action of G on iB(χ). The second condition G makes sure that iB(χ) is a smooth representation. G Proposition 4.8.1. If χ is unitary the iB(χ) is a unitary representation. More generally, G G −1 −1 −1 −1 the contragredient of iB(χ) is iB(χ ) where χ = (χ1 , χ2 ). G G G Proof. Let f1 ∈ iB(χ1, χ2) and f2 ∈ iB(χ1, χ2) then f1f2 ∈ IndB(δ). There is now a canonical G G-invariant linear form IndB(δ) → C. This defines a nonzero G-invariant pairing between G G −1 −1 iB(χ1, χ2) and iB(χ1 , χ2 ). 0 Proposition 4.8.2. Let χ = (χ1, χ2) be a character of A. Except if χ = (χ1, χ2) or 0 G G 0 χ = (χ2, χ1), there is no nonzero intertwining operators from iB(χ) to iB(χ ). G G 0 Proof. Assume that there is a nonzero intertwining operator iB(χ) → iB(χ ). It induces a G G 0−1 non zero G-invariant bilinear form iB(χ) × iB(χ ) → C. We have canonical projections G Λχ : S(G) → iB(χ) defined by Z −1/2 Λχ(φ)(g) = φ(bg)δ (b)χ(b)db B 61 where db is the left invariant measure on B. Similarly, there is a G-equivariant canonical maps G 0−1 Λχ0−1 : S(G) → iB(χ ). It follows that there is a canonical linear form c : S(G × G) → C defined by the formula

c(φ1 ⊗ φ2) = hΛχ(φ1), Λχ0−1 (φ2)i G G 0−1 using the pairing between iB(χ) and iB(χ ). This is a G-invariant with respect to the diagonal action of G by right translation of G on G × G. There exists a unique linear form b : S(G) → C such that 0 c(φ1 ⊗ φ2) = b(φ1 ∗ φ2) 0 −1 where φ2(g) = φ2(g ). Now from the construction we have −1/2 Λχ(lb1 φ1) = δ (b1)χ(b1)Λχ(φ1). Similarly, we have −1/2 0−1 −1 −1 Λχ0 (lb2 φ2) = δ (b2)χ (b2)Λχ0 (φ2). We have 0 0 (4.8.1) (lb1 φ1) ∗ (lb2 φ2) = lb1 rb2 (φ1 ∗ φ2). It follows taht for all φ ∈ S(G), we have −1/2 −1/2 0−1 b(lb1 rb2 φ) = δ (b1)χ(b1)δ (b2)χ (b2)Λ(φ).

We use again the Bruhat decomposition G = Bw1B ∪ B. On each Bruhat double coset there is at most one distribution satisfying the transformation property (4.8.1). On the big cell Bw1B, there is a non zero distribution satisfying (4.8.1) if and only if the restriction of the character −1/2 −1/2 0−1 (4.8.2) (b1, b2) 7→ δ (b1)χ(b1)δ (b2)χ (b2)

to the stabilizer {(b1, b2) | lb1 rb2 w1 = w1}. The stabilizer consists in pairs (a1, a2) with −1/2 −1/2 a1, a2 ∈ A such that a1 = w1a2w1. In that case δ (a1)δ (a2) = 1 so that the restriction 0 of the above character to the stabilizer is trivial if and only if χ = w1χ i.e if χ = (χ1, χ2) 0 then χ = (χ2, χ1). On the small cell B, there is a non zero distribution satisfying (4.8.1) if and only if χ = χ0. Here we have to take the modulus character δ of B in to account. G Proposition 4.8.3. The representation iB(χ) admits an invariant one-dimensional subspace −1 −1 × if and only if χ1χ2 (y) = |y| for all y ∈ F . It admits an invariant one-dimensional −1 × quotient if and only if χ1χ2 (y) = |y| for all y ∈ F . G Proof. Consider one-dimensional representation of G which is a subspace of iB(χ). The commutator group SL(2,F ) acts trivially on the one dimensional subspace. It implies that the restriction of the character δ1/2χ to the one dimensional torus y 0 0 y−1 −1 −1 is trivial. In other words, we have χ1χ2 (y) = |y| . 62 G G −1 Now iB(χ) admits a one-dimensional quotient if and only if its contragredient iB(χ ) admits a one-dimensional subspace. G If iB(G) admits a one-dimensional representation as subspace, then the quotient by this × × subspace is called special representation. For every character χ2 : F → C , we denote σ(χ2) G −1 the special representation given as the irreducible quotient of iB(χ1, χ2) where χ1χ2 (y) = |y|−1.

−1 ±1 Proposition 4.8.4. Except the above mentioned cases χ1χ2 (y) = |y| , the representation G iB(χ) is irreducible. Special representations are irreducible. −1 ±1 G Proof. Except in the cases χ1χ2 (y) = |y| , iB(χ) do not have one-dimensional (so ﬁnite dimensional) subspace or quotient. If it is not irreducible, there exists an exact sequence 0 G 00 0 → V → iB(χ) → V → 0 where both V 0 and V 00 are inﬁnite dimensional. It follows that V 0 and V 00 admits at least one G Whittaker functional after 4.7.9 and therefore iB(χ) admits at least two linearly independent Whittaker functionals which would contradict the following statement G Proposition 4.8.5. For every nontrivial additive character ψ, dim(iB(χ))N,ψ = 1. Proof. The proof is similar to 4.8.2 and is based on the Bruhat decomposition. Theorem 4.8.6. Let V be an irreducible admissible representation of G. If V is a principal G 1/2 1/2 series representation iB(χ) then dim(VN ) = 2 and A acts on VN through δ χ and δ w1(χ). If V is a special representation the dim(VN ) = 1. Otherwise VN = 0 i.e. V is supercuspidal representation.

G Proof. If the irreducible representation V ' iB(χ) is a principal series representation then G G 1/2 V ' iB(w1χ). The isomorphism V ' iB(χ) induces a nonzero A-map VN → δ χ. Similarly, 1/2 we have a nonzero A-map VN → δ w1(χ). This implies that dim(VN ) ≥ 2. If dim(VN > 2) 0 then a character χ ∈/ {χ, w1(χ)} such that there exists a nonzero intertwining operator G G 0 V ' iB(χ) → iB(χ ) which would contradict 4.8.2. G Proposition 4.8.7. The space of K(1)-fixed vectors iB(χ) is of dimension no greater than one. It is of dimension one if and only if χ1 and χ2 are unramified characters. Proof. It follows from the Iwasawa decomposition 4.1.2, a function which is K(1)-right invariant and which transform on the left by the character χ is completely determined by its value at the origin. At the origin, the two conditions can reconciled if and only if χ is trivial G K(1) on B ∩ K in other words if χ is an unramified character. In that case dim iB(χ) = 1. 4.9. Kirillov model. Let ψ : F → C× denote a non trivial additive character. Then all × characters of F must be of the form ψa : x 7→ ψ(ax) for some a ∈ F . Let (π, V ) a smooth irreducible representation of G that admits a ψ-Whittaker functional G Λψ : V → C. For every v ∈ V , the corresponding Whittaker function.Wv ∈ IndN (ψ) is given by g 7→ Λ(π(g)v). The construction of the Kirillov model consist in restricting Whittaker function to the subgroup F × of matrices of the form a 0 a = 0 1 63 Proposition 4.9.1. Assume that V is infinite dimensional. If v 6= 0, then there exists a 0 a ∈ F × so that W 6= 0. v 0 1

× Proof. Assume that Wv(a) = 0 for all a ∈ F . This implies that Λψa (v) = 0 because a 0 v 7→ Λ (π v) ψ 0 1 × is a ψa-Whittaker functional. In other words, the image of v in Va vanishes for every a ∈ F . Here we are considering V as a smooth S(F )-module as in 4.6.4. For every x ∈ F , we have 1 x (4.9.1) π v = v. 0 1 0 0 In fact, if v denote the difference of these two vectors, then the image of v in Va vanishes for every a ∈ F so that v0 = 0. The equation (4.9.1) being established for all x ∈ F , v is a N-invariant vector. By assumption, v is also invariant under an open compact subgroup K and in particular by a K ∩ N − where N − is the group of unipotent lower matrices. Now v is stablized by the SL(2,F ) because this group is generated by any two non trivial unipotent matrices, one upper and the other lower triangular. This implies that V is finite dimensional. × For every v, let φv : F → C denote the function a 0 φ (a) = W . v v 0 1 We denote Kr(V ) the subspace of complex valued functions of F × image of the application v 7→ φv. Since this application is injective, V = Kr(V ) so that Kr(V ) also affords a smooth irreducible representation of G. This is the Kirillov model of V . It is easy to write down the action of upper triangular matrices in Kr(V ) a 0 1 b (4.9.2) π φ (y) = φ (ay) and π φ (y) = ψ(by)φ (y) 0 1 v v 0 1 v v

The equations (4.9.2) impose restrictions on φv associated to a vector v. Since v is ﬁxed by some compact subgroup K, φv must be invariant under the translation by a compact subgroup of F × by the ﬁrst equation. Since v is invariant under N ∩ K, the second equation implies φv(y) = 0 if the character b 7→ ψ(by) is non trivial on N ∩ K, in other words there exists C > 0 such that φ(y) = 0 for all |y| > C.

× Proposition 4.9.2. For every v ∈ V , the function φv : F → C is locally constant and its support is contained in a compact subset of F . Moreover, if v has zero image in the Jacquet module VN , then the function φv has compact support Proof. The ﬁrst statement has just been proved. For the second statement, the kernel V (N) of V 7→ VN is generated by vectors of the form 1 b v − v. 0 1 The associated function φ(y) vanishes for |y| small enough so that ψ(by) = 1. 64 Theorem 4.9.3. Let (π, V ) be an inﬁnite-dimensional irreducible smooth representation of G. Let identify V with its Kirillov model. Then the kernel V (N) of the projection from V ∞ × to its Jacquet module VN is exactly Cc (F ). ∞ × Proof. V (N) is a non-zero B1-invariant subspace of Cc (F ). It is enough to prove that ∞ × Cc (F ) as representation of B1 is irreducible. From the description of Jacquet module of V . we deduce the following proposition.

Proposition 4.9.4. Let us denote by πKr the action of the group a b (4.9.3) B = | a ∈ F ×, b ∈ F 1 0 1 ∞ × ∞ acts on Cc (F ) by the equation (4.9.2). Then (πKr,Cc ) is an irreducible representation. ∞ × Proof. Let φ ∈ Cc (F ). The action of F on φ according to (4.9.2) gives rise to an action of ∞ ∞ the algebra Cc (F ). For every f ∈ Cc (F ), we have ˆ πKr(f)(φ)(y) = f(y)φ(y).

The irreducibility of πKr can be easily deduced from this equation. G −1 Theorem 4.9.5. Let (π, V ) = iB(χ1, χ2) be an irreducible principal series i.e. χ1χ2 (t) is not |y| or |y|−1.

(1) Assume taht χ1 6= χ2. Then the space of Kirillov model of V consists of the function on F × that are locally constant, that vanish for large values of |y| and 1/2 1/2 (4.9.4) φ(y) = c1|y| χ1(y) + c2|y| χ2(y)

for for all |y| small and for some constants c1, c2. (2) Assume that χ1 = χ2 = χ. The space of Kirillov model of V consists of the function on F × that are locally constant, that vanish for large values of |y| and 1/2 1/2 (4.9.5) φ(y) = c1|y| χ(y) + c2val(y)|y| χ(y)

for for all |y| small and for some constants c1, c2. 4.10. Local functional equation. We deﬁne local L factor of an irreducible representation G as follows. If π = iB(χ1, χ2) is a principal series representation, we put −s −1 −s −1 L(s, π) = (1 − α1q ) (1 − α2q )

where αi = χi($) if χi is unramified and αi = 0 if χi is ramified. If π = σ(χ2) is a special −s −1 representation then L(s, π) = (1 − α2q ) where again α2 = χ2($) if χi is unramified and α2 = 0 if χ2 is ramified. Proposition 4.10.1. Let (π, V ) be an infinite dimensional irreducible smooth representation of G. If f is an element in the space of its Kirillov model, consider the interal Z (4.10.1) Z(s, f) = f(y)|y|s−1/2d×y F × This integral is convergent for <(s) >> 0 has meromoprphic continuation to all s ∈ C. There exists a unique polynomial L(π, s)−1 ∈ C[q−s] in the variable q−s with free coefficient one such that for every f ∈ V , there exists p(s, f) such that (4.10.2) Z(s, f) = p(s, f)L(s, π). 65 Moreover, one can choose f0 so that Z(s, f0) = L(s, π) i.e. p(s, f0) = 1.

G Proof. Suppose that (π, V ) = iB(χ) is a principal series with χ = (χ1, χ2). For every function × f : F → C in the Kirillov model of π, there exist c1, c2 ∈ C such that 1/2 1/2 f(y) = c1|y| χ1(y) + c2|y| χ2(y) for all t ∈ F × with |y| small. × Let f0 denote the locally constant on F supported by OF deﬁned by 1/2 1/2 f0(y) = c1t χ1(y) + c2t χ2(y). × Then f − f0 is a locally constant function on F with compact support. The zeta integral ±s Z(s, f − f0) is then a polynomial function on q . The zeta integral Z Z s × s × Z(s, f0) = c1 χ1(y)|y| d y + c2 χ2(y)|y| d y. OF OF × Both summands are absolutely convergent for <(s) >> 0. Assume <(s) >> 0. If χ1 : F → C is a ramiﬁed character then Z s × χ1(y)|y| d y = 0 r $ OF for all integer r. It follows that the integral Z −s × χ1(y)|y| d y = 0. OF

If χ1 is an unramiﬁed character with χ1($) = α1 then we have Z ∞ −s × X n −sn χ1(y)|y| d y = α1 p OF n=0 −s −1 = (1 − α1p ) Thus −s −1 −s −1 Z(s, f0) = c1(1 − α1p ) + c2(1 − α2p ) so that Z(s, f ) 0 = c (1 − α p−s) + c (1 − α p−s) L(s, π) 1 2 2 1 −s −s −1 −s −1 is a polynomial function on p . Here L(s, π) = (1 − α1p ) (1 − α2p ) . Notice that −s −s since α1 6= α2, the scalars c1, c2 can be choosen such that c1(1 − α2p ) + c2(1 − α1p ) = 1. In this case we have

p(s, f0) = 1. In the above argument, we only use the asymptotic expansion of f around 0 based on G the Jacquet module of iB(χ). The same argument works with special representation and supercuspidal representation as well. In the supercuspidal case L(s, π) = 1 and we can choose f0 = 1 × . OF 66 More generally, for all characters ξ : F × → C×, we can deﬁne Z (4.10.3) Z(s, f, ξ) = f(y)χ(y)|y|s−1/2d×y F × as well as the factor L(s, π, χ) so that

(4.10.4) Z(s, f, χ) = p(s, f, χ)L(s, π, χ).

Proposition 4.10.2. Let (π, V ) be an irreducible smooth representation of G and let χ be a character of F ×. Then for all but at most two values of s modulo 2πi/ log(q), the dimension of the space of linear functionals Λ: V → C satisfying y 0 (4.10.5) L π v = χ(y)|y|sL(v) 0 1

is equal to one.

Proof. Let L1,L2 denote two linearly independent linear form satisfying the equation 4.10.5. ∞ × × Since V (N) = Cc (F ) and the space of χ-eigendistribution on F is one-dimensional, L1 and L2 are proportional on V (N). There exist c1, c2 not both zero so that c1L1 +c2L2 factors × through VN . But dim VN ≤ 2, and there are at most two characters of F that appears in VN . If y 7→ χ(y)|y|s is not one of these characters, the space of functionals Λ satisfying (4.10.5) is of dimension not greater than one. By the above proposition, we know that f 7→ pχ(f, s) is a non zero element of this space. Therefore, its dimension is at least one. Theorem 4.10.3. Let (π, V ) be an inﬁnite-dimensional irreducible smooth representation of G with central character ω. Let χ be a character of F ×. We identify V with its Kirillov model. There exists a meromorphic function γ(s, π, ξ, ψ) such that for all f ∈ V we have

−1 −1 (4.10.6) Z(1 − s, π(w1)f, ω χ ) = γ(s, π, χ, ψ)Z(s, f, χ)

where w1 is the matrix

0 1 (4.10.7) w = 1 −1 0

−1 −1 Proof. Both fΛ 7→ L1(f) = Z(s, f, χ) and f 7→ L2(f) = Z(1 − s, π(w1)f, ω χ ) are linear functional on V that satisfy

a 0 (4.10.8) L π f = χ(a)−1|a|−s+1/2L(f). 0 1

First we check this property for L1. Z a 0 s−1/2 × L1(π f) = f(ay)χ(y)|y| d y 0 1 F × = χ(a)−1|a|−s+1/2L(f). 67 Now we will check this property for L2. Z a 0 a 0 −1 −1 −s+1/2 × L2(π f) = π(w1)π f (y)(ω χ )(y)|y| d y 0 1 F × 0 1 Z −1 a 0 −1 −1 −s+1/2 × = ω(a) π π(w1)f (y) ω (y)χ (y)|y| d y F × 0 1 Z −1 −1 −1 −1 −s+1/2 × = (π(w1)f)(a y) ω (a y)χ (y)|y| d y F × −1 −s+1/2 = χ(a) |a| L2(f) For all s except two at complex values, the space of such linear functionals is one dimensional. There exists a proportionality constant γ(s, π, ξ, ψ) between the two linear form which −1 −1 depend holomorphically on s. Both Z(1−s, π(w1)f, ω χ ) and Z(s, f, χ) are meromorphic function on s, then so is γ(s, π, χ, ψ). Theorem 4.10.4. There exists an invertible holomorphic function σ(s, π, ψ) such that L(1 − s, π,˜ χ−1) γ(s, π, χ, ψ) = (s, π, χ, ψ)L(s, π, χ) where π˜ is the contragredient representation of π. Proof. By deﬁnition, we have Z(1 − s, π(w )f, ω−1χ−1) γ(s, π, χ, ψ) = 1 Z(s, f, χ) for all functions f : F × → C in the Kirillov model of π. Let consider the function f˜ : F × → C given by ˜ −1 f(y) = π(w1)f(y)ω (y). We observe that f˜ is an element of the Kirillov model of the contragrendient representation G π˜ of π. If π = iB(χ1, χ2) is an unramiﬁed principal series, there exists c1, c2 ∈ C such that 1/2 π(w1)f(y) = δ (y)(c1χ1(y) + c2χ2(y)) for small |y|. We have ω(y) = χ1χ2(y) so that ˜ 1/2 −1 −1 f(y) = δ (y)(c1χ1 (y) + c2χ2 (y)) G −1 −1 belongs to the Kirillov model of the contragredient representationπ ˜ = iB(χ1 , χ2 ). The same argument works in other cases. In fact f 7→ f˜ induces a bijection from the Kirillov model of π on the Kirillov model ofπ ˜. We have −1 −1 ˜ −1 −1 Z(1 − s, π(w1)f, ω χ ) = p(s, f, χ )L(1 − s, π,˜ χ ) where p(s, f,˜ χ−1) is some polynomial in q±s which is equal one for some f. We have L(1 − s, π,˜ χ−1) γ(s, π, χ, ψ) = (s, π, χ, ψ)L(s, π, χ) with p(s, f,˜ χ−1) (s, π, χ, ψ) = . p(s, f, χ) 68 We exploit the fact the does not depend on f to prove that it is an invertible holomorphic function. We can choose f such that p(s, f, χ) = 1 i.e. is a holomorphic function on s. We can also choose f such that p(s, f,˜ χ−1) = 1 which implies that is inverse to a holomorphic function. So is an invertible holomorphic function. 4.11. Formal Mellin transform.

5. Automorphic representations on adelic groups In this chapter A will denote the ring of adèlesof Q. Let Afin denote the ring of finite fin adèles.We have A = A × R. More generally, if F is a global field, we will denote AF its fin ∞ ring of adèles.We have AF = AF × AF . In this chapter, we will use the letter G for the group GL2.

5.1. Adèlesand idèles. The absolute values of the field of rational numbers Q is either archimedean or non archimedean, thus associated a prime number p or archimedean. Recall −ordp(m)+ordp(n) that the p-adic absolute value of m/n with m, n ∈ Z is |m/n| = p where ordp of an integer is the order of the largest power of p dividing it. While the completion of Q with respect to the archimedean absolute value is the field R of real numbers, the completion Qp of Q with the p-adic absolute value is the field of p-adic numbers. It is the fraction field of the ring Zp of p-adic integers that is the completion of Z with respect to the p-adic absolute value. The ring A of adèlesof Q is the restricted product Yˇ = p × A p∈P Q R whose elements are xA = (xp, x)p∈P with x ∈ R, xP ∈ Qp for all p and xp ∈ Zp for all but fin fin Qˇ finitely many p ∈ P. We also write A = A × R where A = p∈P Qp is called the ring of finite adèles. We equip A and Afin with the Tychonoff product topology which makes it Q ˆ locally compact. The compact subring p∈P Zp is the profinite completion Z of the ring of integers Z. The field of rational numbers Q is contained in A as a discrete subring. Q Q Lemma 5.1.1. We have Q ∩ p∈P Zp = Z and Q\A/ p∈P Zp = R/Z. In particular Q\A is a compact group. Proof. A rational number m/n with (m, n) = 1 that is p-integral for all prime p as n has ˆ fin Q no prime factors. Thus Q ∩ Z = Z. A coset in A / p∈P Zp can be represented by a finite rp collection of fractions (mp/p )p∈S indexed by a finite set S of prime numbers. The sum P rp rp m/n = p∈Σ mp/p satisfies the property that m/n ≡ mp/p for all p. This implies that fin Q Q A = Q + p∈P Zp. It also follows that Q\A/ p∈P Zp = R/Z. The group A× of idèlesof Q is the restricted product

× Yˇ × × = p × A p∈P Q R × × × whose elements are xA = (xp, x)p∈P with x ∈ R , xP ∈ Qp for all p and xp ∈ Zp for all but × × finitely many p ∈ P. We also write A× = Afin × R× where Afin is the group of finite idèles. 69 × Q For xA ∈ A , let us consider the norm |xA| = p∈P |xp|p|x∞|∞ the infinite product is well defined because all but finitely many of its terms is equal to one. Let A1 be the subgroup of elements of norm one in A×. × Q × × × Q × × Lemma 5.1.2. We have Q ∩ p∈P Zp = {±1} and Q \A / p∈P Zp = R+. The group of norm one idèles A1 contains Q× as a cocompact subgroup i.e. the quotient Q×\A1 is a compact group. Proof. A rational number that is p-integral and whose inverse is also p-integral must be ±1. × × For every prime p, there is an isomorphism Qp /Zp = Z given by the p-adic valuation. The fin× Q quotient A / p∈P Zp is given by a collection of integers (rp)p∈P that vanish except for Q rp finitely many of them. Since this class can be represented by the rational number p∈P p , × fin× Q × × Q × × we have Q \A / p∈P Zp = 1. It follows that Q \A / p∈P Zp = R /{±1} that is × isomorphic to R+ by the application x 7→ |x|. In order to prove Q× ⊂ A1, it is enough to prove the product formula Y |x|∞ |x|p = 1 p∈P for all x ∈ Z − {0}. But this follows immediately from the decomposition in prime factors Q rp −rp Q rp x = ± p . We have |x|p = p for all prime p and |x|∞ = p . According to the × × Q × × × 1 Q × isomorphism Q \A / p∈P Zp = R+ above, we have Q \A / p∈P Zp = 1. This implies × 1 the compacity of Q \A .

For every number field F , the ring of adèles AF is defined in a similar manner. An absolute value of F is either archimedean or non archimedean. For an archimedean absolute value v, an infinite place, the completion Fv is either the field of real numbers C or the field of real numbers R. A non archimedean absolute value v, a finite place, is up to normalization is an extension of the p-adic valuation on Q; the v-adic completion Fv is a finite extension of Qp. We will denote by Ov its ring of integers. The ring of adèles AF is the restricted product of all those local fields Fv whose elements are (xv) with xv ∈ Fv for all v and xv ∈ Ov for all but finitely many v. Again, AF = F∞ × AF,fin where F∞ is the product of the completion at infinite places and the ring of finite adèles AF,fin is the restricted product of the completions Q at all finite places. We denote Ofin = v Ov the product over all the finite places v of F of the valuation ring Ov in Fv. Let OF denote the ring of integers of F .

Proposition 5.1.3. The quotient F \AF is a compact group.

Proof. We have F ∩Oﬁn = OF , the ring of integers of F . It follows that F \AF /Oﬁn = OF \F∞. Here F∞ = F ⊗QR is a r-dimensional real vector space that contains OF as a complete lattice. The quotient OF \F∞ is thus compact. × × × × Proposition 5.1.4. AF,1 contains F as a subgroup and the quotient F \AF,1 is a compact group.

× × × Proof. We will see that F \AF,1/Ofin is a compact group. Consider the projection on the finite idèles × × × × × × prfin : F \AF,1/Ofin → F \AF,fin/Ofin. 70 The latter group is the finite by the theorem of finiteness of ideal classes. The kernel of prfin × × × × × × is (F ∩ Ofin)\F∞. We have F ∩ Ofin = OF , the group of units of OF . The compacity of × 1 the quotient OF \F∞ is a formulation of the Dirichlet unit theorem. × × The group F \AF is called the group of idèlesclasses. This is a locally compact group equipped with an absolute value × × + F \AF → R × 1 whose kernel F \AF is a compact group. 5.2. The case of function fields. Let F be the field of rational functions on a smooth projective curve connected C defined over a finite field k. The places of F are the closed points of C. Of course, there is no archimedean place. The ring of adèles AF contains a Q discrete subring F and a compact subring OAF = v∈|C| Ov where Ov is the completion of the local ring OC,v of C at v.

0 1 Proposition 5.2.1. We have F ∩ OAF = H (C, OC ) and F \AF /OAF = H (C, OC ). In particular, they are both finite dimensional k-vector spaces. Proposition 5.2.2. The quotient ×/O× can be identified with the group of divisors of C, AF AF the double quotient F ×\ ×/O× with the group Pic of isomorphism classes of line bundle AF AF C on C. The degree of a line bundle is a homomorphism deg : F ×\ ×/O× → whose kernel, AF AF Z 0 the group PicC of line bundles of degree 0, is finite. × × − deg(x) The absolute value on F \AF is given by |x| = q where q is the cardinal of the base × 1 0 field k. Its kernel is the compact group F \AF that is an extension of the finite group PicC by the compact group O× . The choice of a line bundle of degree one provides a splitting AF 0 PicC = PicC × Z. 5.3. Strong approximation theorem.

fin Theorem 5.3.1. SL2(Q) is dense in SL2(A ). fin Proof. Let M denote the closure of SL2(Q) in SL2(A ). We first prove that M contains fin SL(Qp) embedded as the p-component of SL2(A ). It is enough to prove that M contains the − subgroups N(Qp) and N (Qp) since these subgroups generate SL2(Qp). But this statement derives from the density of Q in Afin. Q Let S be a finite set of primes. It is enough to prove that M contains p∈S SL2(Qp) × Q fin p/∈S SL2(Zp) because by enlarging S, these groups cover SL2(A ). Since M contains already Q SL2(Qp) for p ∈ S, it is enough to prove that M contains the profinite group p/∈S SL2(Zp). Q This is equivalent to prove that M ∩ p/∈S SL2(Zp) maps onto the finite quotient SL2(Z/nZ) for every integer n prime to S. But this follows from the fact SL2(Zp) ⊂ M for all p|n that we already know. fin Corollary 5.3.2. For every compact open subgroup K0 of SL(A ), let us denote Γ0 = SL2(Q) ∩ K0.The embedding of SL2(R) as the infinite component of SL2(A) induces a homeomorphism Γ\SL2(R) → SL2(Q)\SL2(A)/K0. 71 Proof. We first prove that the map

SL2(R) → SL2(Q)\SL2(A)/K0 is surjective. An element of fin SL2(A)/K0 = (SL2(A )/K0) × SL2(R). fin fin fin can be written as x = (x , g∞) with x ∈ SL2(A )/K0 and g∞ ∈ SL2(R). The density fin of SL2(Q) in SL2(A ) implies that there exists γ ∈ SL2(Q) such that whose image in fin fin −1 fin −1 SL2(A )/K0 is x . The transformation formula γ (x , g∞) = (1, γ g∞) implies the fin fin desired surjectivity. Now let γ1, γ2 ∈ SL2(Q) having the same image x ∈ SL2(A )/K0 then γ2 = γ1γ where γ ∈ SL2(Q) ∩ K0 = Γ0. It follows that the map Γ\SL2(R) → SL2(Q)\SL2(A)/K0 is a homeomorphism. Q For every positive integer N let K0(N) = p K0(N)p where K0(N)p be the subgroup of GL2(Zp) of matrix with congruent to an upper triangular matrices modulo N. We have Γ0(N) = SL2(Q) ∩ K0(N). Proposition 5.3.3. We have a homeomorphism

Γ0(N)\SL2(R) ' Z(A)GL2(Q)\GL2(A)/K0(N). Proof. This is the combination of the above corollary and the fact that Q has class number one. 5.4. Automorphic representations and automorphic forms. Proposition 5.4.1. The quotient space G(Q)Z(A)\G(A) has ﬁnite measure. For every unitary character ω : F ×\A× → C×, we consider the space L2(G(F )\G(A), ω) of G(F )-functions φ on G(A), that transform by the character ω with respect to the action of the center Z, and whose module |φ| is a square integrable function on G(Q)Z(A)\G(A). A function φ is said to be cuspidal if Z φ(ng)dn = 0 N(F )\N(A) 2 for all g ∈ G(A). The subspace of cuspidal functions Lcusp(G(F )\G(A), ω) is a closed sub- 2 2 2 space of L (G(F )\G(A), ω). Both L (G(F )\G(A), ω) and Lcusp(G(F )\G(A), ω) are Hilbert representations of G(A).

2 Theorem 5.4.2. The space Lcusp(G(F )\G(A), ω) decomposes as a Hilbert direct sum of irreducible invariant subspaces. Let A(G, ω) denote the space of smooth functions φ : G(A) → C such that (1) φ transforms under the action of Z(A) according to ω, (2) φ is K-finite with respect to any compact subgroup K = KfinK∞ where Kfin is a compact open subgroup of G(Afin) and Kfin is the maximal compact subgroup of SL2(R), 72 (3) φ is Z(U(g))-finite N (4) φ has moderate growth i.e. |φ(g)| ≤ C||g|| for some constant C ∈ R+ and N ∈ N. Such a function φ is called an automorphic form. The K-finiteness implies that there exists fin a compact open subgroup K0 of G(A ). If ω = 1 then φ can be seen as an automorphic function

φ :Γ0\SL2(R) → C. As we have already seen in the framework on automorphic representations on real groups, a L2-automorphic functions that are K-ﬁnite and Z(U(g))-ﬁnite are analytic and will automatically have moderate growth. We will also consider the space

2 Acusp(G, ω) = A(G) ∩ Lcusp(G(F )\G(A), ω) ﬁn of cuspidal automorphic functions. The spaces A(G, ω) and Acusp(G, ω) are smooth G(A )- representations and (g,K∞)-modules.

Deﬁnition 5.4.3. An irreducible admissible G(A)-module is a restricted tensor product M π = πp p with p runs over all places of Q where

(1) (π∞,V∞) is an irreducible admissible (g,K)-module (2) for all finite place p, (πp,Vp) is an irreducible admissible representation of G(Qp), Kp (3) for all but finitely many p, πp is unramified i.e dim(Vp ) = 1. By restricted tensor products we mean the space of vector of the form O v = vp p

Kp where vp ∈ Vp for almost all prime p.

Theorem 5.4.4. Let π be an irreducible G(A)-invariant subspace of L2(G(F )\G(A), ω). The π ∩ A(G, ω) is an admissible G(A)-module in the above sense. 5.5. Fourier expansion and Whittaker models. Theorem 5.5.1. Let (π, V ) be an automorphic cuspidal representation of G with

V ⊂ A0(G(F )\G(A), ω) where ω : F ×\A× → C× is a unitary character of the center. If φ ∈ V and g ∈ G(A), let Z 1 x W (g) = φ g ψ(−x)dx. φ 0 1 F \A

The space WV of functions Wφ is a Whittaker model of π. We have the Fourier expansion X a 0 (5.5.1) φ(g) = W g . φ 0 1 a∈F × 73 Proof. The function f : A → C 1 x f(x) = φ g 0 1 is smooth and periodic with respect to the discrete cocompact subgroup F . It admits the Fourier expansion X f(x) = c(a)ψ(ax) a∈F since F can be identiﬁed with the Pontryagin dual of F \A. The coeﬃcient c(a) is given by the integral Z 1 x c(a) = φ ψ(−ax)dx. 0 1 F \A Since φ is cuspidal c(0) = 0. For a ∈ F ×, we have a 0 Z 1 x a 0 W g = φ g ψ(−x)dx φ 0 1 0 1 0 1 F \A Z a 0 1 a−1x = φ g ψ(−x)dx 0 1 0 1 F \A Z 1 x = φ ψ(−ax)dx. 0 1 F \A = c(a) a 0 where we have used the invariance property of φ with respect to the translation of . 0 1

5.6. Multiplicity one. We will prove the strong multiplicity one theorem of Piatetski- Shapiro. Theorem 5.6.1. Let (π, V ) and (π0,V 0) be two automorphic cuspidal representations of G i.e. 0 V,V ⊂ Acusp(G(F )\G(A), ω) 0 for some unitary character ω of the center. Assume that πv ' πv for all infinite places and for all but finitely many finite places. Then V = V 0.

0 0 Proof. If πv ' πv for all v then they have the same model of Whittaker. The equality V = V follows from (5.5.1). 0 Now we suppose that πv ' πv for all infinite places and for all but finitely many finite L L 0 0 places. Consider their Whittaker model Wh(Vv) and Wh(V ) with Wh(Vv) = Wh(V ) v v v N v for almost all places including the infinite place. We choose Whittaker functions W = v Wv 0 N 0 and W = v Wv as follows 0 (1) W∞ = W∞ is an eigenvector of K∞ 0 0 (2) for all p such that Wh(Vp) = Wh(Vp ) is unramified we choose Wv = Wv being a normalized Kv-invariant vector, 0 (3) for finitely remaining v we choose Wv and Wv such that they have the same Kirillov × function Fv → C which is a smooth compactly supported function. 74 The last condition can be made possible because Kirillov model of any infinite dimensional ∞ × representation contains Cc (Fv ). By 5.5.1 the series X a 0 φ(g) = W g 0 1 a∈F × and X a 0 φ0(g) = W 0 g 0 1 a∈F × are convergent to nonzero function φ, φ0 : G(F )\G(A) → C. By our choice of W and W 0, they agree on B(A). They are both invariant on the right by some K0 where K0 is some fin compact open subgroup of G(A ) and is a eigenvector of K∞ with the same eigenvalue. It follows that φ and φ0 agree over

B(A)K0K∞. The strong approximation theorem implies that

G(F )B(A)K0K∞ = G(A). Since both φ and φ0 are G(F )-invariant on the left, it follows φ = φ0. 0 The irreducible representations V and V sharing a nonzero vector, are the equal. 5.7. Hecke theory from the Jacquet-Langlands point of view. N Theorem 5.7.1. Let π = πv be an irreducible admissible G(A)-module that occurs in Qv Acusp(G, ω). Define L(s, π) = v L(s, πv). Then (1) L(s, π) and L(s, π˜) converge in a right half plane and can be holomorphically continued to C. (2) They are bounded in any finite vertical strip. Q (3) L(s, π) = (s, π)L(1 − s, π˜) with (s, π) = (s, πv, ψv) for any nontrivial additive character ψ : F \A → C. Proof. Let V denote the space of the automorphic cuspidal representation π. For each φ ∈ V , we consider the integral Z a 0 (5.7.1) Z(s, φ) = φ |a|s−1/2d×a. 0 1 F ×\A× Since φ is rapidly decreasing when |a| → ∞, this integral is absolutely convergent for all s ∈ C and defines a holomorphic function in s which is bounded on every vertical strip. The Fourier expansion of φ X a 0 φ(g) = W g φ 0 1 a∈F × allows us to unfold this integral as Z a 0 (5.7.2) I(s, φ) = W |a|s−1/2d×a. φ 0 1 A× 75 If φ is a decomposable vector, its Whittaker function is an infinite product Y Wφ(g) = Wφ,v(gv) v for all g ∈ G(A). Here for all most all v, Wφ,v is the normalised Whittaker function and gv ∈ G(Ov) so that Wφ,v(gv) = 1. For <(s) ≥ 0, (5.7.2) is an Eulerian product Z Y av 0 s−1/2 × Wφ,v |av| d av. × 0 1 v Fv Thus for <(s) ≥ 0, we have Y Z(s, φ) = Z(s, Wφ,v). v

We have Z(s, Wφ,v) = L(s, πv)p(s, WW,φ,v) where p(s, WW,φ,v) is a polynomial which is equal to one if Wφ,v is the normalized unramified Whittaker function. It follows that Y Z(s, φ) = L(s, π) p(s, Wφ,v) v ramified and thus L(s, πv) admit a meromorphic continuation. At the ramified place, it is possible to choose Wφ,v so that p(s, Wφ,v) = 1 which implies that L(s, π) is holomorphic if π is a automorphic cuspidal representation. Using the automorphy of φ, we have another development of Z(φ, s) as Euler product a 0 0 1 a 0 φ = φ 0 1 −1 0 0 1 a−1 0 0 1 = ω(a)φ 0 1 −1 0 Thus Z a−1 0 0 1 Z(φ, s) = φ ω(a)|a|s−1/2d×a × × 0 1 −1 0 F \AF Z a 0 −1 1/2−s × = (π(w1)φ) ω (a)|a| d a × × 0 1 F \AF after the change of variable a 7→ a−1. For <(s) << 0, we have

Y −1 Z(φ, s) = Z(1 − s, ω (π(w1)Wφ,v)) v Y = L(1 − s, π˜v)(s, πv, ψv)p(s, Wφ,v) v

according to the local functional equation. Here (s, πv, ψv) and p(s, Wφ,v) are equal to 1 for almost all v. This implies that L(s, π˜) has meromorphic continuation to all the complex plane and that we have the functional equation L(1 − s, π˜)(s, π) = L(s, π)

where is an exponential function. 76 For all character χ : F ×\A× → C×, the same argument prove that for all automorphic cuspidal representation π the L-function L(s, π, χ) has holomorphic continuation as well as L(1 − s, π,˜ χ−1) and we have the functional equation L(s, π, χ) = (s, π, χ)L(1 − s, π,˜ χ−1). Moreover, the argument can be reversed to prove the converse theorem. Theorem 5.7.2. Let π be an irreducible admissible G(A)-module so that L(s, π, χ) is holomorphic and satisﬁes the above functional equation for all character χ. Then π is automorphic cuspidal.

77 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 → P1 and by assigning t 7→ f, we make F C 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 P points x ∈ X. Its elements are finite linear combinations dixi with di ∈ Z and x ∈ X. P P i The application i dixi 7→ i di defines a homomorphism deg : Div(X) → Z. We denote Div0(X) the group of divisors of degree 0. A divisor D ∈ Div(X) is said to be effective if P D = dixi with di ≥ 0. We will write simply D ≥ 0 if D is effective. i P For every f ∈ F , we have a divisor div(f) = x νx(f)x where νx(f) is the vanishing or νx pole order of f at the point x. If ux is a parameter of X at x then f ∼ ux up to the multiplication by invertible function at x. We have div(f) ∈ Div0(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 divU (f) = P x∈U νx(f)x. We will denote OX the structural sheaf of the the algebraic curve X whose generic fiber is F . For every Zariski open subset U ⊂ X, the sections of OX (U) are regular algebraic functions on U. In other words

Γ(U, OX ) = {f ∈ F | divU (f) ≥ 0}. For every x, the local ring of germs of regular functions at x is

OX,x = {f ∈ F | νx(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 OX,x is called a parameter of X at x. More generally, for every D ∈ Div(X), the sheaf OX (D) is deﬁned by

Γ(U, OX (D)) = {f ∈ F | D|U + divU (f) ≥ 0}. For every x, the group of germs of its sections regular at x is

OX (D)x = {f ∈ F | dx + νx(f) ≥ 0}. 0 0 Its is clear that for every U ⊂ X,H (U, OX (D)) is a H (U, OX )-module and for every x ∈ X, OX (D)x is a OX,x-free module of rank one. This means that for every D ∈ Div(X), OX (D) is a locally free OX -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, OX (D)) = {f ∈ F | D + div(f) ≥ 0}

is a ﬁnite dimensional. If deg(D) < 0, we have Γ(X, OX (D)) = 0. If deg(D) > 2g − 2 where g is the genus of X, we have

dim Γ(X, OX (D)) = 1 − g + deg(D). 78 A.2. Line bundles. This theorem generalizes to any line bundle because every line bundle

L is of the form OX (D). More precisely, let a ∈ L⊗OX F be a meromorphic section of L. For νx(a) every x ∈ X, let νx(a) be the integer so that we have the equivalence relation a ∼ lxux up to multiplication by an invertible unction at x. Here lx is a generator of Lx as OX,x-module P and ux is a generator of the maximal ideal of Ox. Let define D = div(a) = x∈X νx(a)x. Then we have a canonical isomorphism L = OX (D). Over the generic fiber, this is the isomorphism between one-dimensional F -vector spaces F → L ⊗OX F assigning 1 7→ a. Even if any line bundle L on X is of the form L = OX (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 Hi(X, L) are finite dimensional and vanish if i∈ / {0, 1}. We have dim H0(X, L) − dim H1(X, L) = 1 − g + deg(L).

The sheaf of 1-forms ΩX/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, OX ), we have Γ(U, ΩX/k) = ΩA/k where ΩA/k is the A-module of Kahler differentials. Theorem A.2.2 (Serre). There is a canonical non degenerate pairing between H0(X, L) and 1 −1 1 H (X, L ⊗ ΩX/k). In particular, we have a canonical isomorphism H (X, ΩX/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 ux be a local parameter of X at x and uy a local parameter of Y at y. We say that f is étale at y if ux is a local parameter of Y at y, in other words ux and uy 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 étalepoint y ∈ Y is also called a ramified point. There exists an integer ey ey, the ramification index, such that ux ∼ uy 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 X 2gY − 2 = d(2gX − 2) + (ey − 1) y where we sum over all ramification points of Y . ∗ One can pull back a 1-form from X to Y . This defines homomorphism f ΩX → ΩY which which is an injective map whose cokernel is a torsion sheaf ΩY/X supported by the ramified ∗ points. We have deg(f ΩX ) = d(2gX − 2) and deg(ΩY ) = 2gY − 2. It is not difficult to evaluate the length of the cokernel. Let y ∈ Y be a ramified point. We derives from the ey ey−1 relation ux ∼ uy that dux ∼ uy duy which means that the length of the direct factor of ΩY/X supported by y is ey − 1. The Hurwitz theorem follows. A.4. Adelic desctiption. Following Weil, any line bundle admit adelic description as follows. Recall that Fx is the completion of F with respect to the topology defined by the maximal ideal of the local ring OX,x and Ox its ring of integers. In constrast of the local 79 ring OX,x remembers about the curve X because its field of fractions is F , the completed local ring Ox is always isomorphic to the ring of formal series of one variable. Let AF denote the ring of tuples (fx)x∈X with fx ∈ Fx for all x ∈ X and fx ∈ Ox for all but finitely many Q x ∈ X. In particular LA contains both F and x Ox.

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

Lx = L ⊗F Fx is equipped with a Ox-submodule Lx = L ⊗OX Ox which is a free Ox-module of rank one. The adelic data (L, (Lx)x∈X ) is required to satisfy the following condition : for every nonzero element a ∈ L, a is generator of Lx for all but ﬁnitely many x ∈ X. We Q denote LA = L ⊗F AF that contains both L and x Lx. 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

0 Y (A.4.1) H (X, L) = L ∩ Lx x 1 Y (A.4.2) H (X, L) = LA/(L + Lx). x

For each x ∈ X, we have a canonical map ΩX ⊗OX Fx → C given by the residue at x. By Q taking the sum of residue, we have a map LA → C whose restriction to x ΩX ⊗OX Ox vanish by construction, and whose restriction to ΩX,x ⊗OX F vanishes by the residue theorem. This deﬁnes a canonical map H1(X, L) → C that appears in Serre’s theorem.

Appendix B. Fourier transform and the Poisson summation formula See [9, chapter 5] for more details and proofs.

B.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 0, f (2),... are rapidly decreasing, in the sense that (B.1.1) sup |x|k|f (`)(x)| < ∞ for every k, ` ≥ 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 7→ fˆ with Z ∞ fˆ(y) = e−2πixyf(x)dx. −∞

Theorem B.1.1. If f ∈ S(R) then fˆ ∈ 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 d ` yk fˆ(y) dy 80 is the Fourier transform of 1 d k (−2πix)`f(x) (2πi)k dx which is also a Schwartz function. B.2. The Poisson summation formula.

Theorem B.2.1. Let f ∈ S(R) be a Schwartz function. The series ∞ ∞ X X f(n) and fˆ(n). n=−∞ n=−∞ are absolutely convergent. Moreover, we have the equality ∞ ∞ X X f(n) = f(n). n=−∞ n=−∞ Proof. [9, p.154]. 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 > 0 and A > 0 such that A |f(x)| < . 1 + |x|1+ Pn=∞ The series 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 fˆ are sufficiently decreasing. Proposition B.2.2. For any function f that is twice continously derivable and if f, f 0, f (2) have moderate decrease, the Poisson summation formula holds.

Appendix C. Review on Haar measures C.1. Locally compact groups. Let G be a locally compact topological group. We denote the action of G on itself by translation on the left by the formula l(g1)g = g1g. The action by −1 translation on the right is given by r(g1)g = gg1 . These actions induces two actions of G on −1 the space Cc(G) of continuous functions with compact support, given by l(g1)f(g) = f(g1 g) and r(g1)f(g) = f(gg1). By duality, we have action on the left and on the right of G on the space of linear functionals of Cc(G). Proposition C.1.1 (Haar). There exists one and up to a factor only one left-invariant measure dl(G)g and we call it a left-invariant Haar measure.

Obviously, there is also a right invariant measure dr(G) well deﬁned up to a positive factor. Left and right invariant measures are related by the Haar modulus. The right translation r(g1)dl(G) is still a left-invariant measure, so that there is a unique positive constant δG(g1) + such that r(g1)dl(G)g = δG(g1)dl(G)g, and the map g1 7→ δG(g1) deﬁnes a character G → R .

Lemma C.1.2. If dl(G)g is a left invariant measure then δG(g)dl(G)g is a right invariant measure. 81 The group G is said to be unimodular if the Haar modulus is trivial. This is the case for instant if G is its owned derived group [G, G] so that all characters G → R+ ought to be trivial. It is also the case if G is compact because any there is no non trivial compact + subgroup in R . In this case, we will write dg for both dl(G)g and dr(G)g. In the case of Lie group i.e. a topological group equipped with a manifold structure. Here manifold can be deﬁned over F = R, C or a p-adic ﬁeld. Let g be the tangent space of Vdim(G) G at the origin. Given by a non zero linear form g → F where F = R, C or Qp respectively, we have a left (resp. right) invariant volume form ωl(G) (resp. ωr(G)) by left (resp. right) translation. These forms are related by

ωr(G) = det(ad(g))ωl(G) where ad(g) denote the adjoint action. Recall that ad(g) is the derivation of the conjugation action x 7→ gxg−1 of G on itself. We also have densities |ω|l(G) and |ω|r(G) associated to these volume forms. They are of course related by

ωr(G) = | det(ad(g))|ωl(G)

and also the left (right.) invariant measure dl(G)g (resp. dr(G)g).

C.2. Quotient space. Let H be a closed subgroup of a locally compact group G. For all × characters χ : H → C , we consider the space Cc(H\G, χ) consisting of locally constant function f : G → C that satisfy the equation f(hg) = χ(h)f(g) for all h ∈ H and g ∈ G and such that there exists a compact Kf so that f is supported on HKf . Proposition C.2.1. (1) We have a linear surjective map

(C.2.1) Λ : Cc(G) → Cc(H\G, δH ) R that assigns to every f ∈ Cc(G) the function (Λf)(x) = H f(hg)dl(H)h. (2) This application satisﬁes the property Λ(l(h)f) = Λ(f)

and for every continuous linear map Cc(G) → E invariant under l(H) must factorize through Λ. In particular the left invariant measure as a linear functional Cc(G) → C factorizes through Cc(H\G, δH ). (3) If G is unimodular, the induced linear map Cc(H\G, δH ) → C is invariant under the right action of G on H\G. All such G-invariant linear forms are proportional. Proof. The main observation here is that Λ does not commute with the left translation by −1 h1 ∈ H if δH is non trivial. Let us check first the invariant property Λ(l(h1 )f) = Λ(f) for all h1 ∈ H. For all g ∈ G, we have Z −1 Λ(l(h1 )f)(g) = f(h1hg)dl(H)h H Z 0 0 = f(h g)dl(H)h H = Λ(f)(g). 82 For every h1 ∈ H, we have −1 (l(h1 )(Λf))(g) = (Λf)(h1g) Z = f(hh1g)dl(H)h H Z 0 0 = δH (h1) f(h g)dl(H)h H = δH (h1)Λf(g) 0 where we used the change of variables h = hh1 and the transformation rule of measures 0 δH (h1)dl(H)h = dl(H)h. Let f ∈ Cc(G) be a function supported by a compact set Kf . The function Λf is then supported by HKf . Thus Λ defines a linear map Cc(G) → Cc(H\G, δH ). For every x ∈ H\G, there exists an open subset Ux so that the quotient map G → H\G admits a section over ux : Ux → G. We can identify the preimage of Ux in G with H ×ux(Ux). −1 We can also assume Ux is contained in a compact subset of H\G. Let φ ∈ Cc(H\G, δH ) so that f is supported by HKf where Kf is a compact subet of G. Since the image of Kf in H\G is compact, there exists a finite number of x ∈ H\G so that the union of Ux covers the image of Kf . Using the partition of unity, we an assume that φ is supported on H × x(Ux). We can now choose a function f in the form f = fH ⊗ φ so that fH is continuous and R compactly supported and such that H fH (h)dl(H)h = 1. Since Ux is contained in a compact, f is compactly supported in G and we have Λf = φ. We can prove that for every continuous linear map Cc(G) → E that is invariant under l(H) has to factorise through Λ : Cc(G) → Cc(H\G, δH ). In particular the left invariant measure Cc(G) → C factorises through a canonical linear map I : Cc(H\G, δH ) → C. Assume that G is unimodular i.e. dl(G)g = dr(G)g. The straightforward calculation Z I(r(g1)Λ(f)) = f(gg1)dg G Z = f(g0)dg0 G = L(Λ(f)). shows that the map I : Cc(H\G, δH ) → C is invariant under the right action of G We are mainly interested in the case where there exists a compact subgroup K of G so that G = HK. In that case the support condition is automatically satisfied so that we have Cc(H\G, δH ) = C(H\G, δH ). Moreover, the linear form L : C(H\G, δH ) → C can also defined more explicitly as integration over K. Proposition C.2.2. Suppose that there exists a compact subgroup K so that G = HK. Then the restriction to K defines an isomorphism

C(H\G, δH ) → C(H ∩ K\K).

The linear form I : C(H\G, δH ) → C and the integration on K with respect to its invariant measure are proportional with respect to this identiﬁcation.

Proof. Let f ∈ C(H\G, δH ). It derives from G = HK and f(hk) = δH (h)f(k) that f is uniquely determined by its restriction to K. Since K is compact, its closed subgroup H ∩ K 83 is also compact. It follows that the restriction of the character δH to H ∩ K is trivial. Thus the restriction to K defines a linear bijection C(H\G, δH ) → C(H ∩ K\K). The linear form I : C(H\G, δH ) → C defines a linear form C(H ∩ K\K) that is K- invariant on the right. It must be proportional with the linear form C(H ∩ K\K) defined by integration over K.

Appendix D. Operators For more details, see [8, chapter VI].

D.1. Compact operators. Deﬁnition D.1.1. Let X,Y be Banach spaces. A bounded operator T : X → Y is called compact if it takes bounded subset in X into a precompact set in Y i.e. a subset of Y whose closure is compact. In other words, for every bounded sequences {xn} ⊂ X, {T xn} has a convergent subsequence in Y . Important examples of compact operator are integral operator on a compact Haussdorﬀ space, in particular on the unit interval [0, 1]. Let K(x, y) be a continuous function of two variables x, y ∈ [0, 1]. The operator Z 1 (Kφ)(x) = K(x, y)φ(y)dy 0 is a bounded operator on the Banach space C[0, 1] with the uniform norm

||φ||∞ = max φ(x). x∈[0,1] We have indeed

||Kφ||∞ ≤ max K(x, y)||φ||∞. x,y∈[0,1] ∞ Let B1 be the set of φ ∈ C [x, y] with norm ||φ||∞ ≤ 1. The operator K takes B1 into an equicontinuous family. By compacity, for every > 0, there exists δ > 0 so that |K(x, y) − K(x0, y)| < for every y ∈ [0, 1] as long as |x−x0| < δ. This implies |(Kφ)(x)−(Kφ)(x0)| ≤ for every φ ∈ B1. By the Ascoli theorem, a bounded equicontinuous family of functions on [0, 1] is precompact. In other words, K is a compact operator on [0, 1]. Lemma D.1.2. Let S,T ∈ L(X). IF S or T is compact then ST is compact. In other words, compact operators form a left and right ideal of L(X). Proposition D.1.3. Operators of finite rank are compact and a norm limit of operators of finite rank is also compact. Inversely, every compact operator on a separable Hilbert space is the norm limit of a sequence of operators of finite rank. Proof. Balls in finite dimensional vector spaces are compact, and therefore operators of finite rank are compact. Let T : E → F be norm limit of compact operators Tn : E → F . Let xm be a bounded sequence in E. After the extracion of a subsequence, we can assume that for every integers m1, m2 ≥ n, we have |T xm1 − T xm2 | ≤ 1/n. It follows that T xm is a Cauchy sequence which ought be convergent as F is complete. 84 ∞ Let {xi}i=1 be an orthonormal basis of H. It suffices to prove that the finite rank operators n X v 7→ hxi, viT xi i=1 converge to T with respect to the norm topology. It suffices to prove that the norm of the restriction of T to the Hilbert subspace generated by xn+1, xn+2,... tends to zero as n goes to infinity. This derives from the compacity of T . Proposition D.1.4. Let M be a locally compact topological space equipped with a measure, 2 2 and H = L (M). For every K ∈ L (M × M), the operator φ 7→ AK (φ) with Z AK (φ)(x) = K(x, y)φ(y)dy M is a compact operator.

Proof. It follows from the Cauchy-Swartz inequality that ||AK || ≤ ||K||2. Thus the operators AK is continuous. ∞ 2 Let {φi}i=1 be an orthonormal basis for L (M). Then {φi(x)φj(y)} is an orthonormal basis for L2(M × M) so ∞ X K = αijφi(x)φj(y). i,j=1 Consider the kernels n X Kn = αijφi(x)φj(y) i,j=1 2 whose associated operators AKn are of ﬁnite rank. Since Kn → K as n → ∞ in L -norm,

AKn → AK in norm topology as n → ∞. It follows that AK is a compact operator. Theorem D.1.5 (The Fredholm alternative). If A is a compact operator on H, then either (1 − A)−1 exists as bounded operator or Av = v has a nonzero solution. Proof. Let F be a finite rank operator such that ||A−F || < 1. Then the operator 1−(A−F ) P∞ n is invertible and its inverse is the infinite series 1 + n=1(A − F ) . We have ∞ ∞ X X (1 − A)(F (1 + (A − F )n)) = (1 − (A − F ) − F )(1 + (A − F )n) n=1 n=1 ∞ X = 1 − F (1 + (A − F )n) n=1 P∞ n where F (1 + n=1(A − F ) ) is a finite rank operator. It is thus enough to consider the case of finite rank operator instead of compact operators. In this case (1 − A) is invertible if and only if the determinant of the restriction of 1 − A to the range of A, which is finite dimensional, is non zero. This is equivalent to the existence of a nonzero solution of the equation Av = v. Theorem D.1.6 (Analytic Fredholm theorem). Let D be an open connected subset of C. Let A : D → L(H) be an analytic family of compact operators. Assume that there exists z ∈ D so that (1 − A(z))−1 exists as bounded operator. Then (1 − A(z))−1 is meromorphic 85 in D, analytic in the complement D − S of a discrete set S ⊂ D. Its residue at every point z ∈ S is an operator of finite rank. Moreover, at every point z ∈ S, there exists a nonzero solution of the equation A(z)v = v.

Proof. Similar to the aproof of the Fredholm alternative, see [8, p.202]. Theorem D.1.7 (Riesz-Schauder). Let A be a compact operator on a Hilbert space H. There exists a bounded subset σ(A) ⊂ C with no accumulation points but 0 so that (λ − A)−1 exists on C − σ(A). Moreover, every nonzero λ ∈ σ(A) is an eigenvalue of ﬁnite multiplicity.

Appendix E. Abelian L-functions

E.1. Characters and Hecke characters. Let F be a global field. We denote by Fv its × local fields and AF its ring of adèles. We seek a description of characters χ : Fv → C as well as characters of the idèlesclass group F ×\A× → C. Our convention is that characters are continuous homomorphisms with values in C×, and unitary characters are continuous homomorphisms with values in the unit circle C1. + Recall that the local field Fv is equipped with an absolute value x 7→ |x| ∈ R . Its kernel 1 Uv is a compact subgroup of Fv. If Fv = R, then Uv = {±1}. If Fv = C then Uv = C . If × × Fv in non archimedean, then Uv = Ov . There exists a homomorphism Fv → Uv noted by x 7→ uv(x) constructed as follows. If v is archimedean, uv(x) is the unique element of Uv + such that x = uv(x)y with y ∈ R . In the non archimedean case, we need to make a further × choice of a prime element v ∈ Fv . Then uv(x) is the unique element of Uv such that

ordv(x) x = uv(x)v . + Thus, Fv = Uv × R if v is archimedean and Fv = Uv × Z if v is archimedean. × A character χ : Fv → C trivial on Uv is called unramiﬁed. All unramiﬁed character is of the form x 7→ |x|s = es log |x|

for a certain complex number s. This complex number is well defined of Fv is archimedean. In the non-archimedean case, it is only well defined modulo 2πi log |v|. The character x 7→ |x|s is unitary if and only if s is purely imaginary. In the archimedean case, the space of unramified characters is the complex plane C with the imaginary line iR as the space of unitary unramified characters. In the non archimedean, the space of unramified characters is the cylinder C/2πi log |v| with the imaginary circle iR/2πi log |v| as the space of unitary unramified characters. The characters of the compact group Uv is automatically unitary. They form the unitary ˆ ˆ dual Uv of Uv; Uv is a discrete group as Uv is a compact group. If Fv = R, Uv = {±1}, its ˆ 1 ˆ dual is also Uv = {±1}. If Fv = C, Uv = C , its dual is Uv = Z. If Fv is non archimedean, Uv is an inverse limit of finite group, its dual is a direct limit of finite groups. × s Proposition E.1.1. The characters χ : Fv → C are all of the form χ(x) = χu(uv(x))|x| s where χu is a character of the compact group Uv and where |x| is an unramified character, the complex number s is completely determined by χ in the archimedean case but only modulo 2πi log |v| in the non archimedean case. It is unitary if and only if s is purely imaginary. + Proof. The statement follows from the decomposition Fv = Uv × R if v is archimedean and Fv = Uv × Z if v is archimedean. 86 × × × × × A Hecke character is a character χ : F \AF → C . It induces a character χv : Fv → C for each place v so that we have χ = ⊗vχv. Since χ is continuous, χv is unramified for all but finitely many v. A Hecke character χ is called unramified if all its local components χv are unramified. If F is the field of rational functions of a curve C as in 5.2, an unramified Hecke character × is a character χ : PicC → C . With a choice of a line bundle of degree one, we have a 0 s splitting PicC = PicC × Z. In this case χ will also split as χ = χ0 ⊗ |.| where χ0 is a 0 s s − deg(x)s character of the finite group PicC and where |.| is the character x 7→ |x| = q .

E.2. Eigendistributions for local fields. In [3], Kudla gave an exposition of Tate’s thesis following the treatment to Weil in his Bourbaki talk. 0 Let S(Fv) be the space of Schwartz-Bruhat functions on the local field Fv; its dual S(Fv) ∞ is the space of tempered distributions. If v is non archimedean, S(Fv) is the space Cc (Fv) of complexed valued locally constant functions with compact support. The tempered distributions are linear functionals on it. If v is archimedean, S(Fv) is the space of complexed valued smooth functions which, together with all its derivatives, are rapidly decreasing. This is a Fréchet space. The tempered distributions are the continuous linear functionals on it. × The group Fv acts on Fv and induces an action on the space of Schwartz-Bruhat functions 0 × S(Fv) as well as the space of tempered distributions S(Fv) . For every character χ : Fv → × 0 C , let us denote S(Fv) (χ) the space of eigenvectors corresponding to the eigenvalue χ. × × Proposition E.2.1. Assume v non archimedean. For any character χ : Fv → C , we have 0 dim S(Fv) (χ) = 1.

∞ × × Proof. We have an inclusion Cc (Fv ) ,→ S(Fv) from the space of function on F with compact support into the space of Schwartz-Bruhat functions on Fv. By duality, there is an exact sequences 0 0 ∞ 0 0 → S(Fv)0 → S(Fv) → Cc (Fv) → 0 0 where S(Fv)0 is the space of distributions on Fv supported by 0. Since v is non-archimedean, 0 S(Fv)0 is one-dimensional and generated by the δ-distribution δ0. × For evert character χ : Fv → C , this induces a left exact sequences of χ-eigenspaces 0 0 ∞ 0 0 → S(Fv)0(χ) → S(Fv) (χ) → Cc (Fv) (χ). 0 0 We have dim S(Fv)0(χ0) = 1 for the trivial character χ0, and dim S(Fv)0(χ) = 0 for nontrivial ∞ 0 × characters χ 6= χ0. The space Cc (Fv) (χ) is one-dimensional. If d x is a Haar measure × × on Fv , this space is generated by the distribution χ(x)d x. We have the inequality 1 ≤ 0 0 dim S(Fv) (χ0) ≤ 2 and in the second case we have dim S(Fv) (χ) ≤ 1. For nontrivial 0 character, it remains to prove that dim S(Fv) (χ) ≥ 1 by constructing explicitly a nonzero element of its. For this, we will consider two different cases : χ is unramified and χ is ramified. Later on, we will consider separately the case of the trivial character. ∞ × −1 The linear application S(Fv) → Cc (Fv ) mapping f on the function f(x) − f(v x) is × ∞ × Fv -equivariant. By composing with the application Cc (Fv ) → C given by the integration ×x 0 × × against the measure χ(x)d , we get an element S(Fv) (χ) for any character χ : F → C .

This element may or may not be zero. Let’s evaluate it on the function f = 1Ov . The function −1 × x 7→ f(x) − f( x) is the characteristic function 1 × of the maximal compact subgroup O v Ov v × × of F . If χ is unramified, the integration of 1 × against χ(x)d x is one. We defined in this v Ov 87 0 0 case a nonzero element of dim S(Fv) (χ) that we will denote by z0(χ) ∈ S(Fv) (χ). It follows 0 that if χ in unramified and nontrivial dim S(Fv) (χ) = 1. Let us consider now the case of a ramified character. In this case, for every f ∈ S(Fv), the integral Z f(x)χ(x)d×x × n Fv −v Ov n is convergent and independent of n for large n since f is constant on v Ov for n large enough. 0 This defines a nonzero element z0(χ) ∈ S(Fv) (χ) for every ramified character χ. Finally, we consider the case of the trivial character χ0 ... s s By will write z0(s, χ) = z0(χ|.| ) to emphasize on the dependance on the twisting by |.| × when the complex number s varies. For an unitary character χ of Fv , the local zeta integral Z (E.2.1) z(s, χ; f) := f(x)χ(x)|x|sd×x × Fv is absolutely convergent for all f ∈ S(Fv) provided <(s) > 0. In this range, the distribution 0 s f 7→ z(s, χ; f) defines a nonzero element z(s, χ) ∈ S(Fv) (χ|.| ). A direct calculation shows that −1 (E.2.2) z0(s, χ) = Lv(s, χ) z(s, χ) when the right hand side is defined i.e. for all <(s) > 0. This expression implies an analytic continuation of z(s, χ) as L(s.χ)z0(s, χ). × × Proposition E.2.2. Assume v archimedean. For any character χ : Fv → C , we have 0 dim S(Fv) (χ) = 1. ∞ × Proof. As in the non archimedean case, we have an inclusion Cc (Fv ) ,→ S(Fv) from the space of function on F × with compact support into the space of Schwartz-Bruhat functions on Fv. By duality, there is an exact sequences 0 0 ∞ 0 0 → S(Fv)0 → S(Fv) → Cc (Fv) → 0 0 where S(Fv)0 is the space of distributions on Fv supported by 0. In contrast with the non 0 archimedean case, the space of distributions supported by zero S(Fv)0 is no longer one- 0 k dimensional. If v is real, S(Fv)0 is generated by the derivatives D δ0 of the δ-function δ0. If 0 k ¯ l v is complex, S(Fv)0 is generated by D D δ0. The above sequence induces a left exact sequences of χ-eigenspaces 0 0 ∞ 0 0 → S(Fv)0(χ) → S(Fv) (χ) → Cc (Fv) (χ). ∞ 0 × It is easy to see that the space Cc (Fv) (χ) is one-dimensional. If d x is a Haar measure on × × 0 Fv , this space is generated by the distribution χ(x)d x. If v is real, S(Fv)0(χ) 6= 0 only for χ −k 0 is one of the character χk(x) = x for positive integers k for which S(Fv)0(χk) is generated k −k −l by D δ0. We have similar conclusion if v is complex, with the characters χk,l(x) = x x¯ . As in the non archimedean case, we consider the zeta integral Z (E.2.3) z(s, χ; f) := f(x)χ(x)|x|sd×x × Fv that is convergent assuming χ unitary and <(s) > 0. In contrast with the non archimedean case, it is the whole Taylor series of f that accounts for the poles of z(s, χ; f) rather than just the value f(0). The proposition follows from the following two lemmas. 88 In the real case, a unitary character χ : R× → C1 must be either the trivial character or s s − 2 the sign character. In both case, we put L(s, χ) = π Γ( 2 ). In the complex case, a unitary character χ : C → C1 must be of the form χ(z) = znz¯−n. −1 Lemma E.2.3. The distribution z0(s, χ) = L(s, χ) z(s, χ) has an entire analytic continu- 0 ation to the whole s-plane, and for all s, defines a basis vector for the space S(Fv) . Proof. The proof is based from an elementary calculation contained in the following lemma. Lemma E.2.4. (See [2, Lemma 3.1.7]) Suppose that f is a continuous function on [0, 1] which has a uniform convergent Taylor expansion ∞ X k f(x) = akx . k=0 Then Z 1 f(x)xs−1dx 0 has meromorphic continuation to all s, with poles only at the values s = −k. The residue at the pole s = −k is the coefficient ak. 1 Fix a nontrivial additive unitary character ψ : Fv → C . This choice induces an isomor- ˆ 1 phism of Fv with its dual Fv = Hom(Fv, C ); we associate to an element y ∈ Fv the unitary character x 7→ ψ(xy). The Fourier transform Z fˆ(y) = f(x)ψ(xy)dx Fv

of a function f ∈ S(Fv) is well deﬁned and again lies in S(Fv). There is a unique choice ˆ of Haar measure dx that is self-dual with respect to this Fourier transform so that fˆ(x) = f(−x). The Fourier transform on S(Fv) gives rise to the Fourier transform on the dual space 0 S(Fv) of tempered distributions. ˆ −1 Proposition E.2.5. If λ is an χ-eigendistribution, its Fourier transform λ is an χ χ1- eigendistribution where χ1 is the absolute value character : χ1(x) = |x|. Corollary E.2.6. There exists a nonzero constant (s, χ, ψ) such that

zˆ0(1 − s, χ) = (s, χ, ψ)z0(s, χ). This constant, called the local epsilon factor, depends on χ, ψ and holomorphically on s.

For a given function f ∈ S(Fv), the local functional equation can be written as the relation z(1 − s, χ−1, fˆ) = γ(s, χ, ψ)z(s, χ, f) where the gamma factor is defined as L(1 − s, χ−1) γ(s, χ, ψ) = (s, χ, ψ) . L(s, χ) In contrast with the -factor, the γ-factor may have pole or zero. 89 E.3. The global functional equation and the Poisson summation formula. Let F be a global field, AF its ring of adèles.The space S(A) of Schwartz-Bruhat functions on A contains all functions of the form f = ⊗vfv where fv ∈ S(Fv) for all places v and fv = 1Ov for almost all v. The finite linear such factorizable functions are dense on S(A). Again the topological dual S(A)0 consisting on continuous linear functionals on S(A) is the space of tempered distributions. Let λv be a distribution on S(Fv) such that hλv, 1O i = 1 for almost v Q all v, then there exists a tempered distribution λ = ⊗λv such that hλ, fi = vhλv, fvi. × Proposition E.3.1. For all character χ of AF , the space of global χ-eigendistributions 0 0 s S(AF ) (χ) has dimensional one. For any s ∈ C, S(AF ) (χ|.| ) is generated by the standard distribution

z0(s, χ) = ⊗vz0(s, χv). × × For a character χ : F \AF and a function f ∈ S(A), we can define the global zeta integral by Z (E.3.1) z(s, χ, f) = f(x)χ(x)|x|sd×x. A× If f is factorizable f = ⊗vfv, this is a product of the local zeta integrals z(s, χv, fv). Since z (s, χ , f ) = 1 as long as χ is unramified and f = 1 × which happens for v∈ / S where 0 v v v v Ov S is a finite set of places, the convergence of the infinite product z(s, χ, f) boils down to the convergence of the local zeta integrals and the convergence of the partial L-function S Q L (s, χ) = v∈ /S Lv(s, χv). We know that this product is absolutely convergent for <(s) > 1. On this half-plane of convergence, we have

z(s, χ) = Λ(s, χ)z0(s, χ) Q where Λ(s, χ) = v Lv(s, χv) is the complete L-function. We fix a global additive character ψ : k\A → C1. By restricting to local components, we can write ψ = ⊗vψv where ψv is unitary character of Fv. This choice permits us to identify A 1 1 with its dual A = Hom(A, C ) compatibly with the local identifications Fv = Hom(Fv, C ). The global Fourier transform S(A) → S(A) is compatible with the local ones i.e., if f = ⊗vfv ˆ ˆ 0 0 then f = ⊗vfv. It induces a Fourier transform of tempered distributions S(A) → S(A) . Theorem E.3.2. The distribution z(s, χ) defined by the global zeta integral (E.3.1) for <(s) > 1 has a meromorphic continuation to the whole s-plane and satisfies the functional equation zˆ(1 − s, χ−1) = z(s, χ). Proof. Since Schwartz functions have rapid decay at ∞, the integral Z s × z>1(s, χ, f) = f(x)χ(x)|x| d x A×,|x|>1 is convergent for all s. −1 Recall that E.2.6 has the formz ˆ0(1 − s, χv ) = v(s, χv, ψv) = z0(s, χv) where the local epsilon factors v(s, χv, ψv) are equal to one for almost all v. We define the global epsilon Q factor by (s, χ) = v v(s, χv, ψv) and obtain the functional equation −1 zˆ0(1 − s, χ ) = (s, χ)z0(s, χ). 90 By comparing this expression with E.3.2, we get the functional equation of the complete L-function. Corollary E.3.3. Λ(s, χ) = (s, χ)Λ(1 − s, χ−1).

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