1991年2月23日 志村五郎 (著) Introduction to the Arithmetic Theory of Automorphic Functions (Paperback ed.). Princeton University Press. (1971-08-01). 


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.

Goro Shimura is Professor of Mathematics at Princeton University.


 CONTENTS
Preface v Notation and terminology xi List of symbols xii Suggestions to the reader xiv

Chapter 1. Fuchsian groups of the first kind 1 1.1. Transformation groups and quotient spaces 1 1.2. Classification of linear fractional transformations 5 1.3. The topological space f\&* 10 1.4. The modular group SL^Z) 14 1.5. The quotient F\£>* as a Riemann surface 17 1.6. Congruence subgroups of SLS(Z) 20

Chapter 2. Automorphic forms and functions 28 2.1. Definition of automorphic forms and functions 28 2.2. Examples of modular forms and functions 32 2.3. The Riemann-Roch theorem 34 2.4. The divisor of an automorphic form 37 2.5. The measure of T\£ 40 2.6. The dimension of the space of cusp forms 45

Chapter 3. Hecke operators and the zeta-functions associated with modular forms 51 3.1. Definition of the Hecke ring 51 3.2. A formal Dirichlet series with an Euler product 55 3.3. The Hecke ring for a congruence subgroup 65 3.4. Action of double cosets on automorphic forms 73 3.5. Hecke operators and their connection with Fourier coefficients.. 77
3.6. The functional equations of the zeta-functions associated with modular forms
89

Chapter 4. Elliptic curves
4.1. Elliptic curves over an arbitrary
4.2. Elliptic curves over C
4.3. Points of finite order on an elliptic curve and the roots of unity.. 100 4.4. Isogenies and endomorphisms of elliptic curves over C 102
96 field 96 98
 viii CONTENTS *
4.5. Automorphisms of an elliptic curve 106 4.6. Integrality properties of the invariant / 107

Chapter 5. Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves Ill
5.1. Preliminary considerations Ill
5.2. Class field theory in the adelic language 115
5.3. Main theorem of complex multiplication of elliptic curves 117
5.4. Construction of class fields over an imaginary quadratic field .. 121
5.5. Complex multiplication of abelian varieties of higher dimension.. 124

Chapter 6. Modular functions of higher level 133 6.1. Modular functions of level N obtained by division of elliptic
curves
6.2. The field of modular functions of level N rational over Q(e
6.3. A generalization of Galois theory
6.4. The adelization of GL2
6.5. The action of {/ong
6.6. The structure of Aut ®)
6.7. The canonical system of models of r \ £ * for all congruence sub-
groups r of GL,(Q) 152 6.8. An explicit reciprocity-law at the fixed points of GQ+ on £> 157 6.9. The action of an element of GQ with negative determinant 163

Chapter 7. Zeta-functions of algebraic curves and abelian varieties — 167 7.1. Definition of the zeta-functions of algebraic curves and abelian
varieties; the aim of this chapter 167 7.2. Algebraic correspondences on algebraic curves 168 7.3. Modular correspondences on the curves Vs 172 7.4. Congruence relations for modular correspondences 176 7.5. Zeta-functions of Vs and the factors of the jacobian variety of Vs 179
7.6. I-adic representations ,
J7.7. Construction of class fields over real quadratic
7.8. The zeta-function of an abelian variety of CM-type 7.9. Supplementary remarks
189 fields 197 211 220

Chapter 8. The cohomology group associated with cusp forms 223 8.1. Cohomology groups of Fuchsian groups 223 8.2. The correspondence between cusp forms and cohomology classes 230 8.3. Action of double cosets on the cohomology group 236 8.4. The complex torus associated with the space of cusp forms — 239
ait(/JV
133 ).. 136 141 143 146 149

CONTENTS ix
Chapter 9. Arithmetic Fuchsian groups 241 9.1. Unit groups of simple algebras 241 9.2. Fuchsian groups obtained from quaternion algebras 243
Appendix 253 References 260 Index 265 Errata 269



https://pdfs.semanticscholar.org/fde9/26fb26a59fd05fec87ca0e4447a3f5250294.pdf
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志村五郎 (著)  Shimura, Goro 書籍 英語版


志村五郎 (著)   Automorphic Functions and Number Theory. Lecture Notes in Mathematics. 54 (Paperback ed.). Springer. (1968).


志村五郎 (著) Introduction to the Arithmetic Theory of Automorphic Functions (Paperback ed.). Princeton University Press. (1971-08-01). 


志村五郎 (著)  Euler Products and Eisenstein Series. CBMS Regional Conference Series in Mathematics (Paperback ed.). American Mathematical Society. (1997-07-01). 


志村五郎 (著)  Abelian Varieties with Complex Multiplication and Modular Functions (Hardcover ed.). Princeton University Press. (1997-12-08).


志村五郎 (著)  Arithmeticity in the Theory of Automorphic Forms. Mathematical Surveys and Monographs (Paperback ed.). American Mathematical Society. (2000-08-22). 


志村五郎 (著)  Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups. Mathematical Surveys and Monographs (Hardcover ed.). American Mathematical Society. (2004-03-01). 


志村五郎 (著) Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Paperback ed.). Springer New York. (2009-12-28). 



志村五郎 (著)  Arithmetic of Quadratic Forms. Springer Monographs in Mathematics (Hardcover ed.). Springer. (2010-07-15). 

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論文集

Collected Papers. I: 1954-1965 (Hardcover ed.). Springer. (2002). 

Collected Papers. II: 1967-1977 (Hardcover ed.). Springer. (2002). 

Collected Papers. III: 1978-1988 (Hardcover ed.). Springer. (2003). 

Collected Papers. IV: 1989-2001 (Hardcover ed.). Springer. (2003). 

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