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18.785 Notes Contents 1 Introduction 4 1.1 What is an automorphic form? . 4 1.2 A rough definition of automorphic forms on Lie groups . 5 1.3 Specializing to G = SL(2; R)....................... 5 1.4 Goals for the course . 7 1.5 Recommended Reading . 7 2 Automorphic forms from elliptic functions 8 2.1 Elliptic Functions . 8 2.2 Constructing elliptic functions . 9 2.3 Examples of Automorphic Forms: Eisenstein Series . 14 2.4 The Fourier expansion of G2k ...................... 17 2.5 The j-function and elliptic curves . 19 3 The geometry of the upper half plane 19 3.1 The topological space ΓnH ........................ 20 3.2 Discrete subgroups of SL(2; R) ..................... 22 3.3 Arithmetic subgroups of SL(2; Q).................... 23 3.4 Linear fractional transformations . 24 3.5 Example: the structure of SL(2; Z)................... 27 3.6 Fundamental domains . 28 3.7 ΓnH∗ as a topological space . 31 3.8 ΓnH∗ as a Riemann surface . 34 3.9 A few basics about compact Riemann surfaces . 35 3.10 The genus of X(Γ) . 37 4 Automorphic Forms for Fuchsian Groups 40 4.1 A general definition of classical automorphic forms . 40 4.2 Dimensions of spaces of modular forms . 42 4.3 The Riemann-Roch theorem . 43 4.4 Proof of dimension formulas . 44 4.5 Modular forms as sections of line bundles . 46 4.6 Poincar´eSeries . 48 4.7 Fourier coefficients of Poincar´eseries . 50 4.8 The Hilbert space of cusp forms . 54 4.9 Basic estimates for Kloosterman sums . 56 4.10 The size of Fourier coefficients for general cusp forms . 60 1 5 L-functions associated to cusp forms 63 5.1 The Mellin transform . 63 5.2 Weil's converse theorem . 69 6 Hecke Operators 73 6.1 Initial Motivation . 73 6.2 Hecke operators on lattices . 75 6.3 Explicit formulas for Hecke operators . 79 6.4 A geometric definition of Hecke operators . 82 6.5 Hecke operators as correspondences . 87 6.6 Brief remarks on Hecke operators for Fuchsian groups . 88 7 Modularity of elliptic curves 90 7.1 The modular equation for Γ0(N). 90 7.2 The canonical model for X0(N) over Q . 93 7.3 The moduli problem and X0(N)..................... 97 7.4 Hecke Correspondences for Y0(N).................... 99 7.5 The Frobenius Map . 101 7.6 Eichler-Shimura theory . 103 7.7 Zeta functions of curves over finite fields . 105 7.8 Hasse-Weil L-functions of a curve over Q . 107 7.9 Eichler-Shimura theory in genus one . 108 8 Defining automorphic forms on groups 111 8.1 Eigenfunctions of ∆ on H { Maass forms . 111 8.2 Automorphic forms on groups { rough version . 117 8.3 Basic Lie theory for SL(2; R) . 118 8.4 K-finite and Z-finite functions . 120 8.5 Distributions and convolution of functions . 121 8.6 A convolution identity for Z-finite, K-finite functions . 126 8.7 Fundamental domains again - Siegel sets . 132 8.8 Growth conditions . 135 8.9 Definition and first properties of an automorphic form . 136 9 Finite dimensionality of automorphic forms 138 9.1 Constant term estimates and cuspidality . 138 9.2 Finite dimensionality of automorphic forms . 142 2 10 Eisenstein series 147 10.1 Definition and initial convergence . 147 10.2 Constant terms of Eisenstein series . 156 10.3 Outline of Proof of Analytic Continuation . 160 10.4 The meromorphic continuation principle . 167 10.5 Continuation consequences: functional equations, etc. 171 10.6 Spectral decomposition of L2(ΓnG) . 173 11 Applications 176 11.1 Review of representation theory . 176 11.2 Representations of SL(2; R) . 177 3 These are (not terribly original) notes for the lectures. For further reading, see the bibliography at the end of Section 1. 1 Introduction 1.1 What is an automorphic form? An automorphic form is a generalization of a certain class of periodic functions. To understand their definition, we begin with the simpler example of periodic functions on the real line. We consider functions f : R ! C which are (for simplicity) periodic with period 1. That is, f(x + n) = f(x) for all x 2 R and all n 2 Z We want to use differential calculus to study them, so we use a translation invariant measure. In this case, its the familiar Lebesgue measure for R which we'll just denote by the usual dx. Generally speaking, such a translation invariant measure on a topological group is referred to as a \Haar measure." A very powerful tool for studying such functions is harmonic analysis. That is, we want to find an orthogonal system with which we may express any (reasonably nice) periodic functions (e.g. integrable functions on R=Z) { this is often referred to as a \complete orthogonal system." Since orthogonal systems are, by assumption, countable we may denote its members rather suggestively by en(x). Then (after normalizing to obtain an orthonormal basis) we may expand f as follows: X Z 1 f(x) = an en(x); where an = f(x)en(x) dx: 0 n2Z As most of you probably already know, one such choice for this orthogonal system 2πinx is to set en(x) = e for each n 2 Z. But why this choice? For inspiration, we can go back to the original problem that motivated Fourier himself. He was seeking a general solution for the heat equation in a thin metal plate. It was known that if the heat source was expressible as a sinusoidal wave, then the solution was similarly expressible as a sinusoidal wave. Fourier's idea was to use a superposition of these waves to attack the problem for an arbitrary function as heat source. In short, a good basis for the solution to his problem came from the eigenfunctions of a differential operator! Which differential operator? In this case, there's a particularly natural choice { the Laplace operator ∆ associated to our metric dx. In general, this can be defined 4 on any Riemannian manifold by ∆(f) = div grad f. In the coordinate x, this is just d2 dx2 for our example. Indeed, the spectral theorem for compact, self-adjoint operators guarantees that the collection of eigenfunctions will give an orthogonal basis. (The word \spectrum" for an operator T here means the set of elements λ such that T − λI is not invertible. This notion will play an especially important role for us in the course.) To the extent that this basis is good for solving the problem at hand, Fourier analysis can be an extremely useful tool. 1.2 A rough definition of automorphic forms on Lie groups To generalize from the setting above, we let X be a locally compact space with a discontinuous group action by a discrete group Γ. Then an automorphic function f : X ! C is just a function invariant with respect to this action: f(γx) = f(x) for all γ in Γ. Recall that a discontinuous action of a discrete group Γ on a topological space X means that for any point x in X, there exists a neighborhood Ux of x such that γ(Ux) \ Ux = ; for all non-trivial γ in Γ. An important special case is when X = G=K, where G is a locally compact group, and K is a closed subgroup. For example, if G is a Lie group, then this quotient is a Riemannian manifold which admits a differential calculus. In general, a deep theorem from functional analysis asserts that all locally compact groups have a (unique up to constant left- or right-) Haar measure. Then an automorphic form is a simultaneous eigenfunction of the algebra D of invariant differential operators on X. This algebra includes the Laplace-Beltrami operator. Typically, we allow for a more general transformation law for these functions: f(γx) = j(γ; x)f(x) where j may consist of differential factors and multiplier systems. (More on those in later lectures). For now, we content ourselves with an example of such a j. Again with x as a coordinate on the manifold X and suppose d(γx) = c(γ; x)d(x). Then setting j(γ; x) = c(γ; x)−1, we obtain all 1-forms on ΓnG=K that are eigenfunctions of D. 1.3 Specializing to G = SL(2; R) For us, our main example will be G = SL(2; R) and K = SO(2; R). Remember, that SL(2; R) = fγ 2 MatR(2 × 2) j det(γ) = 1g; 5 T cos θ − sin θ SO(2; R) = fγ 2 Mat (2×2) j γ γ = I; det(γ) = 1g = θ 2 [0; 2π) : R sin θ cos θ Then the resulting space X = G=K is isomorphic to the complex upper half-plane H = fz 2 C j =(z) > 0g. This isomorphism can be seen from the fact that G acts transitively on the point i in the upper half plane by linear fractional transformation: a b az + b γ = : z ! ; c d cz + d and the stabilizer of i is K. For the discrete group Γ, we often will take SL(2; Z) or a finite index subgroup. The most important class of subgroups are called \congruence subgroups." They are subgroups which contain Γ(n) = fγ 2 SL(2; Z) j γ ≡ I (mod n)g for some positive integer n. To finish the definition, we need only note that the algebra of invariant differential operators is one dimensional, generated by the Laplacian for H. Here the invariant measure (in terms of Lebesgue measure dx and dy) is dµ = y−2dxdy with associated Laplacian @2 @2 @ @ ∆ = y2 + = −(z − z)2 @x2 @y2 @z @z where, as usual, @ 1 @ @ @ 1 @ @ = − i ; = + i : @z 2 @x @y @z 2 @x @y @ The presence of the @z on the right in ∆ explains why \classical automorphic forms" are defined as complex analytic functions f : H! C which satisfy a transformation law of the form: a b f(γz) = (cz + d)kf(z) for all γ = 2 Γ c d for some fixed integer k.
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