Introduction to the Arithmetic Theory of Automorphic Functions (Publications of the Mathematical Society of Japan, Vol. 11)

The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles. INTRODUCTION TO THE ARITHMETIC THEORY OF AUTOMORPHIC FUNCTIONS......Page 1 Title Page......Page 3 Copyright Page......Page 4 Preface......Page 6 Contents......Page 8 Notation and Terminology......Page 12 List of Symbols......Page 13 Suggestions to the Reader......Page 15 1.1. Transformation groups and quotient spaces......Page 16 1.2. Classification of linear fractional transformations......Page 20 1.3. The topological space Γ\H*......Page 25 1.4. The modular group SL 2 (Z)......Page 29 1.5. The quotient Γ\H* as a Riemann surface......Page 32 1.6. Congruence subgroups of SL 2 (Z)......Page 35 2.1. Definition of automorphic forms and functions......Page 43 2.2. Examples of modular forms and functions......Page 47 2.3. The Riemann–Roch theorem......Page 49 2.4. The divisor of an automorphic form......Page 52 2.5. The measure of Γ\H......Page 55 2.6. The dimension of the space of cusp forms......Page 60 3.1. Definition of the Hecke ring......Page 66 3.2. A formal Dirichlet series with an Euler product......Page 70 3.3. The Hecke ring for a congruence subgroup......Page 80 3.4. Action of double cosets on automorphic forms......Page 88 3.5. Hecke operators and their connection with Fourier coefficients......Page 92 3.6. The functional equations of the zeta-functions associated with modular forms......Page 104 4.1. Elliptic curves over an arbitrary field......Page 111 4.2. Elliptic curves over C......Page 113 4.3. Points of finite order on an elliptic curve and the roots of unity......Page 115 4.4. Isogenies and endomorphisms of elliptic curves over C......Page 117 4.5. Automorphisms of an elliptic curve......Page 121 4.6. Integrality properties of the invariant J......Page 122 5.1. Preliminary considerations......Page 126 5.2. Class field theory in the adelic language......Page 130 5.3. Main theorem of complex multiplication of elliptic curves......Page 132 5.4. Construction of class fields over an imaginary quadratic field......Page 136 A. Algebraic preliminaries......Page 139 B. Abelian varieties with many complex multiplications......Page 141 C. Main theorem......Page 144 A. The functions f^i a (z)......Page 148 B. The field generated by the points of finite order on an elliptic curve......Page 150 6.2. The field of modular functions of level N rational over Q(e^(2πi/N))......Page 151 6.3. A generalization of Galois theory......Page 156 6.4. The adelization of GL 2......Page 158 6.5. The action of U on F......Page 161 6.6. The structure of Aut(F)......Page 164 6.7. The canonical system of models of Γ\H* for all congruence subgroups Γ of GL 2 (Q)......Page 167 6.8. An explicit reciprocity-law at the fixed points of G Q+ on H......Page 172 6.9. The action of an element of G Q with negative determinant......Page 178 7.1. Definition of the zeta-functions of algebraic curves and abelian varieties; the aim of this chapter......Page 182 7.2. Algebraic correspondences on algebraic curves......Page 183 7.3. Modular correspondences on the curves V S......Page 187 7.4. Congruence relations for modular correspondences......Page 191 7.5. Zeta-functions of V S and the factors of the jacobian variety of V S......Page 194 7.6. l-adic representations......Page 204 7.7. Construction of class fields over real quadratic fields......Page 212 7.8. The zeta-function of an abelian variety of CM-type......Page 226 A. Change of model and the field of definition......Page 235 C. The Euler factors for the primes where the variety has bad reduction......Page 236 8.1. Cohomology groups of Fuchsian groups......Page 238 8.2. The correspondence between cusp forms and cohomology classes......Page 245 8.3. Action of double cosets on the cohomology group......Page 251 8.4. The complex torus associated with the space of cusp forms......Page 254 9.1. Unit groups of simple algebras......Page 256 9.2. Fuchsian groups obtained from quaternion algebras......Page 258 Appendix......Page 268 References......Page 275 Index......Page 280 Errata......Page 284 Back Cover......Page 287