**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: $84.00

Individual Member Price: $67.20

**Electronic ISBN: 978-1-4704-1342-2
Product Code: SURV/115.S.E**

List Price: $84.00

Individual Member Price: $67.20

#### Supplemental Materials

# Advanced Analytic Number Theory: L-Functions

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*Carlos 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