Theta Constants, Riemann Surfaces and the Modular Group

Transcription

1 Theta Constants, Riemann Surfaces and the Modular Group An Introduction with Applications to Uniformization Theorems, Partition Identities and Combinatorial Number Theory Hershel M. Farkas Irwin Kra Graduate Studies in Mathematics Volume 37 American Mathematical Society Providence, Rhode Island

2 Contents Introduction xv Chapter 1. The modular group and elliptic function theory 1 1. Mobius transformations 2 2. Riemann surfaces 4 3. Kleinian groups Generalities The situation of interest 8 4. The elliptic paradise The family of tori The algebraic curve associated to a torus Invariants for tori Tori with symmetries Congruent numbers The plumbing construction Teichmiiller and moduli spaces for tori Fiber spaces - the Teichmiiller curve Hyperbolic version of elliptic function theory Fuchsian representation Symmetries of once punctured tori The modular group 43 vn

3 viii Contents 5.4. Geometric interpretations The period of a punctured torus The function of degree two on the once punctured torus The quasi-fuchsian representation Subgroups of the modular group Basic properties Poincare metric on simply connected domains Fundamental domains The principal congruence subgroups T(k) Adjoining translations: The subgroups G(k) The Hecke subgroups r o (fc) Structure of T(k, k) A two parameter family of groups A geometric test for primality 68 Chapter 2. Theta functions with characteristics Theta functions and theta constants Definitions and basic properties The transformation formula More transformation formulae Characteristics Classes of characteristics Integral classes of characteristics Rational classes of characteristics Invariant classes for T(k) Punctures on M 2 /T(k) and the classes X o (k) The classes in X o (k) Invariant quadruples Towers ' Punctures and characteristics A correspondence Branching More invariant classes Invariant classes for G(k) 107

4 Contents ix 4.2. Characterization of G(k) The surface M 2 /G(k) Invariant classes for T o (k) More homomorphisms Elliptic function theory revisited Function theory on a torus Projective embeddings of the family of tori Conformal mappings of rectangles and Picard's theorem Reality conditions Hyperbolicity and Picard's theorem Spaces of iv-th order ^-functions The Jacobi triple product identity The triple product identity The quintuple product identity 143 Chapter 3. Function theory for the modular group F and its 147 subgroups 1. Automorphic forms and functions Two Banach spaces Poincare series Relative Poincare series Projections to the surface Factors of automorphy Multiplicative (/-forms Residues Weierstrass points for subspaces of A(H 2, G, e) Automorphic forms constructed from theta constants The order of automorphic forms at cusps - Fourier series expansions at zoo Automorphic forms for T(k) Meromorphic automorphic functions for T(k) Evaluation of automorphic functions at cusps Automorphic forms and functions for G(k) Automorphic forms and functions for T o (k) The structure of L 0 A g (H 2, F) and =0 A+{W 2, T) 179

5 X Contents , Some special cases (k' = 1) k = l k = 2 k = 3 k = A k = 5 fc = 6 Primitive invariant automorphic forms An index 4 subgroup of T(k) for even k A Hilbert space of modified theta constants Projective representation of Aut H 2 /r(a;) More Hilbert spaces of modified theta constants Orders of automorphic forms at cusps Calculations via F 0 (A;) The general case The field of meromorphic functions on M 2 /F(k) Functions of small degree G(/c)-invariant functions Generators for the function field IC(T(k)) Projective representations Some special cases (k' = k) k = 3 k = 5 The function field for H 2 /F(7) The projective embedding of H 2 /r(7) k = 11 A; = 13 fc = 9 The function field of H 2 /r(a:) over M 2 /T Equations that are satisfied by the embedding The residue theorem The algorithm Three term identities Examnles of enuations

6 Contents xi 11. Some special cases (restricted characteristics) Characteristics with m' = k Characteristics with m = k Ratios 263 Chapter 4. Theta constant identities Dimension considerations The septuple product identity Further generalizations Uniformization considerations Elliptic functions as quotients of AT-th order theta functions The Jacobi quartic and derivative formula revisited More identities - revisited More identities - new results More first order applications Some modular equations Identities which arise from modular forms Multiplicative meromorphic forms Cusp forms for T Some special results for the primes 5 and Ramanujan's r-function Identities among infinite products Identities via logarithmic differentiation Averaging automorphic forms The groups G(k) 312 Chapter 5. Partition theory: Ramanujan congruences Calculations of P N {n) Some preliminaries F(p, g)-invariant functions Calculation of divisor of TJ(N-) Coset representatives Generalities on constructions of F o (fc)-invariant functions The basic problems 345

7 xii Contents Some generalities Constructions of (group) F o (A:)-mvariant functions The direct construction Averaging T(k n, fc)-invariant functions Bases for JC(T 0 (k))o and IC(T o (k)) oo Precomposing with Ak Partition identities Production of constant functions The Frobenius automorphism Constant functions Congruences Functions Fk, n,n for negative N Functions F^^-N of small degree Averaging operators Automorphisms of K.(T o (k)) Other linear maps Modular equations k = 2 fc = 3 k = 5 k = 7 k = 13 The ideal of partition identities Examples: Calculations for small k k = 2 fc = 3 k = b k = 7 fe = 11 A; = 13 fc = 4 k = 6 The higher level Ramanujan congruences The level two and three congruences for small primes

8 Contents xiii The level n congruences for the prime The level n congruences for the prime The level two congruences for the prime Taylor series expansions for infinite products 430 Chapter 6. Identities related to partition functions Some more identities related to covering maps fc = k = k = k = k = The j-function and generalizations of the discriminant A Congruences for the Laurent coefficients of the j-function Averaging f k Completion of the proof of Theorem 3.6 for k = Proof of Theorem 3.6 for k = 11, n = A further analysis of the k 2 case 461 Chapter 7. Combinatorial and number theoretic applications Generalities on partitions Euler series and some old identities Partitions and sums of divisors Lambert series Identities among partitions A curious property of A curious property of A curious property of Partitions, divisors, and sums of triangular numbers Sums of 4 squares A remarkable formula Weighted sums of triangular numbers Counting points on conic sections Continued fractions and partitions 499

9 xiv Contents 6. Return to theta functions 504 Bibliography 511 Bibliographical Notes 513 Index. 527