Mathematical Surveys and Monographs
Volume: 115;
2005;
291 pp;
Softcover
MSC: Primary 11; 22;
Print ISBN: 978-0-8218-4266-9
Product Code: SURV/115.S
List Price: $90.00
AMS Member Price: $72.00
MAA Member Price: $81.00
Electronic ISBN: 978-1-4704-1342-2
Product Code: SURV/115.S.E
List Price: $84.00
AMS Member Price: $67.20
MAA Member Price: $75.60
Supplemental Materials
Advanced Analytic Number Theory: L-Functions
Share this pageCarlos Julio Moreno
Since the pioneering work of Euler, Dirichlet, and Riemann, the
analytic properties of L-functions have been used to study the distribution of
prime numbers. With the advent of the Langlands Program, L-functions have
assumed a greater role in the study of the interplay between Diophantine
questions about primes and representation theoretic properties of Galois
representations.
This book provides a complete introduction to the most significant class of
L-functions: the Artin-Hecke L-functions associated to finite-dimensional
representations of Weil groups and to automorphic L-functions of principal type
on the general linear group. In addition to establishing functional equations,
growth estimates, and non-vanishing theorems, a thorough presentation of the
explicit formulas of Riemann type in the context of Artin-Hecke and automorphic
L-functions is also given.
The survey is aimed at mathematicians and graduate students who want to
learn about the modern analytic theory of L-functions and their applications in
number theory and in the theory of automorphic representations. The
requirements for a profitable study of this monograph are a knowledge of basic
number theory and the rudiments of abstract harmonic analysis on locally
compact abelian groups.
Readership
Graduate students and research mathematicians interested in analytic number theory.
Table of Contents
Table of Contents
Advanced Analytic Number Theory: L-Functions
- Contents v6 free
- Preface vii8 free
- Advice to the Reader ix10 free
- Introduction xiii14 free
- Chapter I. HECKE L-FUNCTIONS 122 free
- Chapter II. ARTIN-HECKE L-FUNCTIONS 7394
- 1. Variations on Brauer's Theorem 7394
- 2. Characteristic Polynomials and Integral Formulas 80101
- 3. Representations of Weil Groups 84105
- 4. Relative Local Weil Groups 87108
- 5. Relative Global Weil Groups 88109
- 6. Non-Archimedean Local L-factors 88109
- 7. Conductors of Representations (The Herbrand Distribution) 89110
- 8. Archimedean Local L-Factors 93114
- 9. Decomposition Groups at Infinity 94115
- 10. The Gamma Factors (Non-Abelian Case) 95116
- 11. Local Properties of L-Functions 97118
- 12. Global Properties of L-Functions 99120
- 13. Artin-Hecke L-Functions (Functional Equation) 100121
- 14. The Root Number 102123
- 15. The Local Langlands Correspondence for GL(n) 103124
- 16. The Principle of Functoriality 109130
- Notes 113134
- Chapter III. ANALYTIC PROPERTIES OF L-FUNCTIONS 115136
- Chapter IV. THE EXPLICIT FORMULAS 143164
- Chapter V. BOUNDS ON DISCRIMINANTS AND CONDUCTORS 177198
- Chapter VI. NON-VANISHING THEOREMS 203224
- Appendix A. THE LOCAL THEORY OF ROOT NUMBERS: A Survey 269290
- Bibliography 283304
- Index 291312 free