DOCSLIB.ORG

Automorphic Forms on GL(2)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS Contents 1. Introduction and motivation 2 2. Representation Theoretic Notions 3 2.1. Preliminaries on adeles, groups and representations 3 2.2. Some categories of GF -modules and HF -modules 3 2.3. Contragredient representation 8 3. Classification of local representations: F nonarchimedean 9 3.1. (π; V ) considered as a K-module 10 3.2. A new formulation of smoothness and admissibility 10 3.3. Irreducibility 11 3.4. Induced representations 13 3.5. Supercuspidal representations and the Jacquet module 14 3.6. Matrix coefficients 17 3.7. Corollaries to Theorem 3.6.1 20 3.8. Composition series of B(χ1; χ2) 22 3.9. Intertwining operators 26 3.10. Special representations 36 3.11. Unramified representations and spherical functions 39 3.12. Unitary representations 44 3.13. Whittaker and Kirillov models 45 3.14. L-factors attached to local representations 57 4. Classification of local representations: F archimedean 72 4.1. Two easier problems 72 4.2. From G-modules to (g;K)-modules 73 4.3. An approach to studying (gl2;O(2; R))-modules 77 4.4. Composition series of induced representations 78 4.5. Hecke algebra for GL2(R) 81 4.6. Existence and uniqueness of Whittaker models 82 4.7. Explicit Whittaker functions and the local functional equation 84 4.8. GL2(C) 88 5. Global theory: automorphic forms and representations 88 5.1. Restricted products 88 5.2. The space of automorphic forms 89 5.3. Spectral decomposition and multiplicity one 90 5.4. Archimedean parameters 92 5.5. Classical forms on H 93 5.6. Hecke theory and the converse theorem 94 Date: Spring 2008. 1 2 AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS References 98 These notes were derived primarily from a course taught by Yannan Qiu at the University of Wisconsin in Spring of 2008. Briefly, the course was an introduction to the book of Jacquet and Langlands on Automorphic forms on GL2. I am greatly in debt to Yannan for teaching the class, and to Rob Rhoades who typed most of the original lecture notes. Additional details beyond Rob's notes (including solutions to many of the exercises) are included, and some additions and restructuring of the material has been made. The main reference for the course (and these notes) is Jacquet and Langlands book [5]. Some other references are Bump's book [2] and Jacquet's book [3]. Bump's book is easier to read but the real material is in [5]. Jacquet's book develops theory of GLn automorphic forms. 1. Introduction and motivation This course will be about L-functions and automorphic forms. There are two sorts of L-functions. (1) Artin L-functions: Suppose E=F is a finite extension of number fields and Gal(E=F ) acts on a finite dimensional vector space over C by ρ. Then we get L(s; ρ). (2) Automorphic L-function: Let G be a reductive group over F and π an automorphic representation. Then we get L(s; π). The Langlands Philosophy says that every Artin L-function is automorphic. More precisely, if L(s; ρ) is the Artin L-function corresponding to a non-trivial irreducible finite dimensional Galois representation, then there exists π, a cuspidal automorphic representation of GLn, such that L(s; ρ) = L(s; π). Remark. The full Langlands conjectures says more than just what is above, but they predict that irreducible ρ correspond to cuspidal π. The Langlands philosopy is useful because there are many analytic techniques that can be used to study automorphic L-functions. In particular, they are known to have holomorphic continuation and other such properties. However, many con- jectures (including Artin's conjecture) of number theory would be true if the same properties could be shown to hold for Artin L-functions. Remark. Artin's conjecture is that Artin L-functions are analytic (i.e. holomorphic on the entire complex plane.) It is known that they admit a meromorphic continua- tion, but it appears that the best way to prove holomorphicity is via the Langlands philosophy. The truth of the Langlands philosophy has been shown in the \GL1 case." The 1-dimensional Artin L-functions arise from characters ρ : Gal(E=F ) ! C×. By class field theory, we have ab 0 × × × Gal(E=F ) ! Gal(E=F ) = Gal(E =F ) ' AF =F Nm(AE0 ); 0 for some field E . We construct the automorphic representation of GL1 over F to be × × × × χ : AF =F Nm(AE0 ) ! C : AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS 3 × (Note that GL1(AF ) = AF .) It is true that L(s; ρ) = L(s; χ). The study of L(s; χ) can be found in Tate's thesis[7]. As in the example above, given a Galois representation, the first goal to proving the truth of Langlands philosopy would be to associate to it an automorphic rep- resentation. Then, one would want to show that the corresponding L-functions are the same. The primary goal of this course will be to explain what automorphic L-functions and automorphic representations are. 2. Representation Theoretic Notions 2.1. Preliminaries on adeles, groups and representations. General Set Up: Let G be a group over a global field F (which will typically be Q.) Possible examples include GLn, Sp2m, O(n). In the case of Sp2m we mean that the skew symmetric form is defined over F . To such a group, the notion of an automorphic representation can be defined. We take G = GL2. Then the adelic points of G can be viewed as the restricted product Y∗ G(AF ) = G(Fv) v places of F with respect to the subgroups Kv = GL2(OFv ). That is, (gv) 2 G(AF ) if and only 1 if gv 2 Kv for almost all v . Kν is a maximal compact subgroup of G(Fν ). This is a generalization of the adeles × Y × AF = Fv : v Here, the restricted product is taken with respect to the valuation rings OFν ⊂ Fv. Example 2.1.1. Y∗ × = × × ×: AQ R Qp p prime × Note that GL1(Zp) = Zp : Exercise 2.1.2. Show that GL2(Zp) is a maximal compact subgroup of GL2(Qp). It is a fact that if π is an irreducible automorphic representation of GL2(AF ) then it is the restricted tensor product π = ⊗πv, where πv are irreducible representations Q of GL2(Fv). Attached to each πv is an L-factor L(s; πv), and L(s; π) = L(s; πv) converges absolutely when s 0, and it has mermomorphic continuation. We do not describe what the restricted tensor product means at this time, but instead begin by classifying all local representations πv. We often treat the archimedean and nonarchimedean cases separately. 2.2. Some categories of GF -modules and HF -modules. In this section, we consider representations of GF = GL2(F ), where F is a non-archimedean local field. 4 F is a topological field so GF inherits the subspace topology from M2(F ) = F , and it is totally disconnected. A representation (π; V ) of GF is a complex vector space V together with an action of GF denoted π(g). In other words, V is a GF -module. We may also refer to the representation as π (when V is clear from context), or we may just say V is a GF -module (when the action π is clear from context.) 1i.e for all but finitely many. 4 AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS 2.2.1. Finite dimensional representations. Usually V will be infinite dimensional, but, in this section, suppose V is finite dimensional. Proposition 2.2.1. If (π; V ) is a continuous representation of GF , then ker(π) must contain an open subgroup. For example, GL2(OF ) is an open subgroup. × × Proof. We first prove the GL1 case. Say π = χ : Qp ! GL1(C) = C , this is the case that V is one dimensional. So the preimage of 1 equals ker(χ). Let R be a small open set around 1 in C, then χ−1(R) ⊇ N, N an open normal subgroup and χ(N) ⊂ R. Since N is a group, χ(N) must be a subgroup of C×. For R sufficiently small, the only such group is f1g. Hence, χ(N) = f1g, and N ⊆ ker(χ). This proof readily generalizes to GLn for any n. It is a fact from complex Lie theory, that any open neighborhood of the identity generates the connected component of the identity in the group. The topologies of C and Qp are not compatible, and that is what makes this × possible. C has an archimedean topology and Qp is totally disconnected. Remark. An open problem is to understand finite dimensional p-adic vector space representations which are much more complicated. 2.2.2. Smoothness and admissibility. Proposition 2.2.1 is not true in general when V is infinite dimensional. In the spirit of the proposition, and to obtain more manageable representations, we will restrict ourselves to the class of representations satisfying one or more of the following conditions. (A) (π; V ) is such that for any vector v 2 V , fg 2 GF j π(g)v = vg contains an open subgroup. (B) For all N ⊂ GF open, the set fv 2 V : Nv = vg is finite dimensional If (π; V ) satisfies (A), we call π smooth, and if it satisfies by (A) and (B), we say it is admissible. There is a good theory of admissible representations because it often reduces to that of finite groups where one can use orthogonality conditions and Schur's lemma.
Recommended publications
  • Automorphic Forms on GL(2)

    Automorphic Forms on GL(2)

    Automorphic Forms on GL(2) Herve´ Jacquet and Robert P. Langlands Formerly appeared as volume #114 in the Springer Lecture Notes in Mathematics, 1970, pp. 1-548 Chapter 1 i Table of Contents Introduction ...................................ii Chapter I: Local Theory ..............................1 § 1. Weil representations . 1 § 2. Representations of GL(2,F ) in the non•archimedean case . 12 § 3. The principal series for non•archimedean fields . 46 § 4. Examples of absolutely cuspidal representations . 62 § 5. Representations of GL(2, R) ........................ 77 § 6. Representation of GL(2, C) . 111 § 7. Characters . 121 § 8. Odds and ends . 139 Chapter II: Global Theory ............................152 § 9. The global Hecke algebra . 152 §10. Automorphic forms . 163 §11. Hecke theory . 176 §12. Some extraordinary representations . 203 Chapter III: Quaternion Algebras . 216 §13. Zeta•functions for M(2,F ) . 216 §14. Automorphic forms and quaternion algebras . 239 §15. Some orthogonality relations . 247 §16. An application of the Selberg trace formula . 260 Chapter 1 ii Introduction Two of the best known of Hecke’s achievements are his theory of L•functions with grossen•¨ charakter, which are Dirichlet series which can be represented by Euler products, and his theory of the Euler products, associated to automorphic forms on GL(2). Since a grossencharakter¨ is an automorphic form on GL(1) one is tempted to ask if the Euler products associated to automorphic forms on GL(2) play a role in the theory of numbers similar to that played by the L•functions with grossencharakter.¨ In particular do they bear the same relation to the Artin L•functions associated to two•dimensional representations of a Galois group as the Hecke L•functions bear to the Artin L•functions associated to one•dimensional representations? Although we cannot answer the question definitively one of the principal purposes of these notes is to provide some evidence that the answer is affirmative.
  • Automorphic L-Functions

    Automorphic L-Functions

    Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 27-61 AUTOMORPHIC L-FUNCTIONS A. BOREL This paper is mainly devoted to the L-functions attached by Langlands [35] to an irreducible admissible automorphic representation re of a reductive group G over a global field k and to local and global problems pertaining to them. In the context of this Institute, it is meant to be complementary to various seminars, in particular to the GL2-seminars, and to stress the general case. We shall therefore start directly with the latter, and refer for background and motivation to other seminars, or to some expository articles on this topic in general [3] or on some aspects of it [7], [14], [15], [23]. The representation re is a tensor product re = @,re, over the places of k, where re, is an irreducible admissible representation of G(k,) [11]. Accordingly the L-func­ tions associated to re will be Euler products of local factors associated to the 7r:.'s. The definition of those uses the notion of the L-group LG of, or associated to, G. This is the subject matter of Chapter I, whose presentation has been much influ­ enced by a letter of Deligne to the author. The L-function will then be an Euler product L(s, re, r) assigned to re and to a finite dimensional representation r of LG. (If G = GL"' then the L-group is essentially GLn(C), and we may tacitly take for r the standard representation r n of GLn(C), so that the discussion of GLn can be carried out without any explicit mention of the L-group, as is done in the first six sections of [3].) The local L- ands-factors are defined at all places where G and 1T: are "unramified" in a suitable sense, a condition which excludes at most finitely many places.
  • Automorphic Forms for Some Even Unimodular Lattices

    Automorphic Forms for Some Even Unimodular Lattices

    AUTOMORPHIC FORMS FOR SOME EVEN UNIMODULAR LATTICES NEIL DUMMIGAN AND DAN FRETWELL Abstract. We lookp at genera of even unimodular lattices of rank 12p over the ring of integers of Q( 5) and of rank 8 over the ring of integers of Q( 3), us- ing Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find in- stances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank 12 over the Eisenstein integers, even and unimodular over Z, we prove a conjecture of Hentschel, Krieg and Nebe, identifying a certain linear combination of theta series as an Hermitian Ikeda lift, and we prove that another is an Hermitian Miyawaki lift. 1. Introduction Nebe and Venkov [54] looked at formal linear combinations of the 24 Niemeier lattices, which represent classes in the genus of even, unimodular, Euclidean lattices of rank 24. They found a set of 24 eigenvectors for the action of an adjacency oper- ator for Kneser 2-neighbours, with distinct integer eigenvalues. This is equivalent to computing a set of Hecke eigenforms in a space of scalar-valued modular forms for a definite orthogonal group O24. They conjectured the degrees gi in which the (gi) Siegel theta series Θ (vi) of these eigenvectors are first non-vanishing, and proved them in 22 out of the 24 cases. (gi) Ikeda [37, §7] identified Θ (vi) in terms of Ikeda lifts and Miyawaki lifts, in 20 out of the 24 cases, exploiting his integral construction of Miyawaki lifts.
  • REPRESENTATION THEORY WEEK 7 1. Characters of GL Kand Sn A

    REPRESENTATION THEORY WEEK 7 1. Characters of GL Kand Sn A

    REPRESENTATION THEORY WEEK 7 1. Characters of GLk and Sn A character of an irreducible representation of GLk is a polynomial function con- stant on every conjugacy class. Since the set of diagonalizable matrices is dense in GLk, a character is defined by its values on the subgroup of diagonal matrices in GLk. Thus, one can consider a character as a polynomial function of x1,...,xk. Moreover, a character is a symmetric polynomial of x1,...,xk as the matrices diag (x1,...,xk) and diag xs(1),...,xs(k) are conjugate for any s ∈ Sk. For example, the character of the standard representation in E is equal to x1 + ⊗n n ··· + xk and the character of E is equal to (x1 + ··· + xk) . Let λ = (λ1,...,λk) be such that λ1 ≥ λ2 ≥ ···≥ λk ≥ 0. Let Dλ denote the λj determinant of the k × k-matrix whose i, j entry equals xi . It is clear that Dλ is a skew-symmetric polynomial of x1,...,xk. If ρ = (k − 1,..., 1, 0) then Dρ = i≤j (xi − xj) is the well known Vandermonde determinant. Let Q Dλ+ρ Sλ = . Dρ It is easy to see that Sλ is a symmetric polynomial of x1,...,xk. It is called a Schur λ1 λk polynomial. The leading monomial of Sλ is the x ...xk (if one orders monomials lexicographically) and therefore it is not hard to show that Sλ form a basis in the ring of symmetric polynomials of x1,...,xk. Theorem 1.1. The character of Wλ equals to Sλ. I do not include a proof of this Theorem since it uses beautiful but hard combina- toric.
  • Matrix Coefficients and Linearization Formulas for SL(2)

    Matrix Coefficients and Linearization Formulas for SL(2)

    Matrix Coefficients and Linearization Formulas for SL(2) Robert W. Donley, Jr. (Queensborough Community College) November 17, 2017 Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 1 / 43 Goals of Talk 1 Review of last talk 2 Special Functions 3 Matrix Coefficients 4 Physics Background 5 Matrix calculator for cm;n;k (i; j) (Vanishing of cm;n;k (i; j) at certain parameters) Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 2 / 43 References 1 Andrews, Askey, and Roy, Special Functions (big red book) 2 Vilenkin, Special Functions and the Theory of Group Representations (big purple book) 3 Beiser, Concepts of Modern Physics, 4th edition 4 Donley and Kim, "A rational theory of Clebsch-Gordan coefficients,” preprint. Available on arXiv Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 3 / 43 Review of Last Talk X = SL(2; C)=T n ≥ 0 : V (2n) highest weight space for highest weight 2n, dim(V (2n)) = 2n + 1 ∼ X C[SL(2; C)=T ] = V (2n) n2N T X C[SL(2; C)=T ] = C f2n n2N f2n is called a zonal spherical function of type 2n: That is, T · f2n = f2n: Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 4 / 43 Linearization Formula 1) Weight 0 : t · (f2m f2n) = (t · f2m)(t · f2n) = f2m f2n min(m;n) ∼ P 2) f2m f2n 2 V (2m) ⊗ V (2n) = V (2m + 2n − 2k) k=0 (Clebsch-Gordan decomposition) That is, f2m f2n is also spherical and a finite sum of zonal spherical functions.
  • An Introduction to Lie Groups and Lie Algebras

    An Introduction to Lie Groups and Lie Algebras

    This page intentionally left blank CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 113 EDITORIAL BOARD b. bollobás, w. fulton, a. katok, f. kirwan, p. sarnak, b. simon, b. totaro An Introduction to Lie Groups and Lie Algebras With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups, to the theory of root systems, and classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras. CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: http://www.cambridge.org/series/ sSeries.asp?code=CSAM Already published 60 M. P. Brodmann & R. Y. Sharp Local cohomology 61 J. D. Dixon et al. Analytic pro-p groups 62 R. Stanley Enumerative combinatorics II 63 R. M. Dudley Uniform central limit theorems 64 J.
  • Introduction to L-Functions II: Automorphic L-Functions

    Introduction to L-Functions II: Automorphic L-Functions

    Introduction to L-functions II: Automorphic L-functions References: - D. Bump, Automorphic Forms and Repre- sentations. - J. Cogdell, Notes on L-functions for GL(n) - S. Gelbart and F. Shahidi, Analytic Prop- erties of Automorphic L-functions. 1 First lecture: Tate’s thesis, which develop the theory of L- functions for Hecke characters (automorphic forms of GL(1)). These are degree 1 L-functions, and Tate’s thesis gives an elegant proof that they are “nice”. Today: Higher degree L-functions, which are associated to automorphic forms of GL(n) for general n. Goals: (i) Define the L-function L(s, π) associated to an automorphic representation π. (ii) Discuss ways of showing that L(s, π) is “nice”, following the praradigm of Tate’s the- sis. 2 The group G = GL(n) over F F = number field. Some subgroups of G: ∼ (i) Z = Gm = the center of G; (ii) B = Borel subgroup of upper triangular matrices = T · U; (iii) T = maximal torus of of diagonal elements ∼ n = (Gm) ; (iv) U = unipotent radical of B = upper trian- gular unipotent matrices; (v) For each finite v, Kv = GLn(Ov) = maximal compact subgroup. 3 Automorphic Forms on G An automorphic form on G is a function f : G(F )\G(A) −→ C satisfying some smoothness and finiteness con- ditions. The space of such functions is denoted by A(G). The group G(A) acts on A(G) by right trans- lation: (g · f)(h)= f(hg). An irreducible subquotient π of A(G) is an au- tomorphic representation. 4 Cusp Forms Let P = M ·N be any parabolic subgroup of G.
  • Math.GR] 21 Jul 2017 01-00357

    Math.GR] 21 Jul 2017 01-00357

    TWISTED BURNSIDE-FROBENIUS THEORY FOR ENDOMORPHISMS OF POLYCYCLIC GROUPS ALEXANDER FEL’SHTYN AND EVGENIJ TROITSKY Abstract. Let R(ϕ) be the number of ϕ-conjugacy (or Reidemeister) classes of an endo- morphism ϕ of a group G. We prove for several classes of groups (including polycyclic) that the number R(ϕ) is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space G, when R(ϕ) < ∞. Applying the result to iterations of ϕ we obtain Gauss congruences for Reidemeister numbers. b In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with R∞ property. Introduction The Reidemeister number or ϕ-conjugacy number of an endomorphism ϕ of a group G is the number of its Reidemeister or ϕ-conjugacy classes, defined by the equivalence g ∼ xgϕ(x−1). The interest in twisted conjugacy relations has its origins, in particular, in the Nielsen- Reidemeister fixed point theory (see, e.g. [29, 30, 6]), in Arthur-Selberg theory (see, e.g. [42, 1]), Algebraic Geometry (see, e.g. [27]), and Galois cohomology (see, e.g. [41]). In representation theory twisted conjugacy probably occurs first in [23] (see, e.g. [44, 37]). An important problem in the field is to identify the Reidemeister numbers with numbers of fixed points on an appropriate space in a way respecting iterations. This opens possibility of obtaining congruences for Reidemeister numbers and other important information. For the role of the above “appropriate space” typically some versions of unitary dual can be taken.
  • Automorphic Forms on Os+2,2(R)+ and Generalized Kac

    Automorphic Forms on Os+2,2(R)+ and Generalized Kac

    + Automorphic forms on Os+2,2(R) and generalized Kac-Moody algebras. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨urich, 1994), 744–752, Birkh¨auser,Basel, 1995. Richard E. Borcherds * Department of Mathematics, University of California at Berkeley, CA 94720-3840, USA. e-mail: [email protected] We discuss how modular forms and automorphic forms can be written as infinite products, and how some of these infinite products appear in the theory of generalized Kac-Moody algebras. This paper is based on my talk at the ICM, and is an exposition of [B5]. 1. Product formulas for modular forms. We will start off by listing some apparently random and unrelated facts about modular forms, which will begin to make sense in a page or two. A modular form of level 1 and weight k is a holomorphic function f on the upper half plane {τ ∈ C|=(τ) > 0} such that f((aτ + b)/(cτ + d)) = (cτ + d)kf(τ) ab for cd ∈ SL2(Z) that is “holomorphic at the cusps”. Recall that the ring of modular forms of level 1 is P n P n generated by E4(τ) = 1+240 n>0 σ3(n)q of weight 4 and E6(τ) = 1−504 n>0 σ5(n)q of weight 6, where 2πiτ P k 3 2 q = e and σk(n) = d|n d . There is a well known product formula for ∆(τ) = (E4(τ) − E6(τ) )/1728 Y ∆(τ) = q (1 − qn)24 n>0 due to Jacobi. This suggests that we could try to write other modular forms, for example E4 or E6, as infinite products.
  • 18.785 Notes

    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 .
  • Computing with Algebraic Automorphic Forms

    Computing with Algebraic Automorphic Forms

    Computing with algebraic automorphic forms David Loeffler Abstract These are the notes of a five-lecture course presented at the Computations with modular forms summer school, aimed at graduate students in number theory and related areas. Sections 1–4 give a sketch of the theory of reductive algebraic groups over Q, and of Gross’s purely algebraic definition of automorphic forms in the special case when G(R) is compact. Sections 5–9 describe how these automor- phic forms can be explicitly computed, concentrating on the case of definite unitary groups; and sections 10 and 11 describe how to relate the results of these computa- tions to Galois representations, and present some examples where the corresponding Galois representations can be identified, giving illustrations of various instances of Langlands’ functoriality conjectures. 1 A user’s guide to reductive groups Let F be a field. An algebraic group over F is a group object in the category of alge- braic varieties over F. More concretely, it is an algebraic variety G over F together with: • a “multiplication” map G × G ! G, • an “inversion” map G ! G, • an “identity”, a distinguished point of G(F). These are required to satisfy the obvious analogues of the usual group axioms. Then G(A) is a group, for any F-algebra A. It’s clear that we can define a morphism of algebraic groups over F in an obvious way, giving us a category of algebraic groups over F. Some important examples of algebraic groups include: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK, e-mail: d.a.
  • In Honor of Bertram Kostant

    In Honor of Bertram Kostant

    Progress in Mathematics Volume 123 Series Editors J. Oesterle A. Weinstein Lie Theory and Geometry In Honor of Bertram Kostant Jean-Luc Brylinski Ranee Brylinski Victor Guillemin Victor Kac Editors Springer Science+Business Media, LLC Jean-Luc Brylinski Ranee Brylinski Department of Mathematics Department of Mathematics Penn State University Penn State University University Park, PA 16802 University Park, PA 16802 Victor Guillemin Victor Kac Department of Mathematics Department of Mathematics MIT MIT Cambridge, MA 02139 Cambridge, MA 02139 Library of Congress Cataloging In-Publication Data Lie theory and geometry : in honor of Bertram Kostant / Jean-Luc Brylinski... [et al.], editors. p. cm. - (Progress in mathematics ; v. 123) Invited papers, some originated at a symposium held at MIT in May 1993. Includes bibliographical references. ISBN 978-1-4612-6685-3 ISBN 978-1-4612-0261-5 (eBook) DOI 10.1007/978-1-4612-0261-5 1. Lie groups. 2. Geometry. I. Kostant, Bertram. II. Brylinski, J.-L. (Jean-Luc) III. Series: Progress in mathematics : vol. 123. QA387.L54 1994 94-32297 512'55~dc20 CIP Printed on acid-free paper © Springer Science+Business Media New York 1994 Originally published by Birkhäuser Boston in 1994 Softcover reprint of the hardcover 1st edition 1994 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy• ing, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Springer Science+Business Media, LLC, for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A.