L-Functions

Description

This book provides an accessible introduction to the theory of L-functions, emphasising their central role in number theory and their direct applications to key results. Designed to be elementary, it offers readers a clear pathway into the subject, starting from minimal background. It describes several important classes of L-functions — Riemann and Dedekind zeta functions, Dirichlet L-functions, and Hecke L-functions (for characters with finite image) — by showing how they are all special cases of the construction, due to Artin, of the L-function of a Galois representation. The analytic properties of abelian L-functions are presented in detail, including the full content of Tate's thesis, which establishes analytic continuation and functional equations via harmonic analysis. General Hecke L-functions are also discussed, using the modern perspective of idèles and adèles to connect their analytic theory with the representation-theoretic approach of Artin's L-functions. A distinguishing feature of this book is its accessibility: while largely avoiding arithmetic geometry, it provides introductions to both algebraic number theory and key aspects of representation theory. This approach ensures that the material is accessible to both beginning graduate students and advanced undergraduates. Applications play a central role throughout, highlighting how L-functions underpin significant results in number theory. The book provides complete proofs of the prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, Chebotarev's density theorem, and the analytic class number formula, demonstrating the power of the theory in solving classical problems. It serves as an ideal introduction for advanced undergraduates and beginning graduate students and can also be a useful reference for preparing a course on the subject. Preface References Contents 1 What Is an L-function? 1.1 The Riemann ζ Function 1.2 Dedekind ζ Functions 1.3 Dirichlet L-functions 1.4 The Selberg Class 1.5 Automorphic Versus Galois: A Tale of Two (Classes of) L-functions Problems References 2 The Prime Number Theorem 2.1 The Riemann–von Mangoldt Exact Formula Problems References 3 Review of Algebraic Number Theory 3.1 Structure of the Ring of Integers 3.2 Unique Factorisation of Ideals 3.3 Splitting of Primes 3.4 Galois Action on the Primes 3.5 Frobenius Elements 3.6 Dirichlet's Unit Theorem and the Regulator 3.7 The Class Group 3.8 Completed ζ Functions: The Local Factors at Infinity Problems Reference 4 A Primer of Representation Theory 4.1 Basic Definitions 4.2 Constructions on Representations 4.2.1 Direct Sum 4.2.2 Dual Representation 4.2.3 Tensor Products and Homomorphisms 4.3 Complete Reducibility 4.4 Character Theory 4.4.1 Further Orthogonality Relations 4.5 Further Operations on Representations Problems Reference 5 The L-function of a Complex Galois Representation 5.1 Independence of the Choice of 𝔓 5.2 Convergence of the Euler Product 5.3 The Riemann and Dedekind ζ Functions, and Dirichlet's L-functions, Are Artin L-functions 5.4 The Formalism of Artin L-functions 5.5 Artin's Conjecture on Analytic Continuation 5.6 Factorisation of the Dedekind ζ-function Problems References 6 Dirichlet's Theorem on Arithmetic Progressions 6.1 Pontryagin Duality: Finite Case 6.2 Densities 6.3 Factorisation of the Cyclotomic Dedekind ζ Function 6.4 Infinitely Many Primes in Arithmetic Progressions 6.5 The Philosophy of Special Values 6.5.1 Further Special Values Problems References 7 The Chebotarev Density Theorem 7.1 Analytic Proof 7.2 Algebraic Proof Problems References 8 The Haar Measure 8.1 Preliminaries 8.2 Haar Measure: Existence 8.3 Haar Measure: Uniqueness (up to Constants) Problems References 9 Abstract Fourier Analysis 9.1 Pontryagin Duality: General Case 9.2 The Abstract Fourier Transform References 10 Review of Local Fields Problems References 11 Restricted Direct Products 11.1 Abstract Group Theory 11.2 Topological Groups 11.3 (Quasi-)Characters of a Restricted Product 11.4 Measure Theory 11.5 Fourier Analysis Problems 12 The Local Theory 12.1 The Additive Group 12.1.1 Choice of Haar Measure 12.2 The Multiplicative Group 12.2.1 Choice of Haar Measure 12.3 Local Zeta Functions I: The General Functional Equation 12.4 Local Zeta Functions II: Computation of the Local Factors 12.4.1 Real Case 12.4.1.1 Conventions 12.4.1.2 Equivalence Classes of Quasi-Characters 12.4.1.3 Choice of f 12.4.1.4 Fourier Transforms 12.4.1.5 The ζ-Functions 12.4.1.6 The Function ρ(c) 12.4.2 Complex Case 12.4.2.1 Conventions 12.4.2.2 Equivalence Classes of Quasi-Characters 12.4.2.3 Choice of f 12.4.2.4 Fourier Transforms 12.4.2.5 The ζ-Functions 12.4.2.6 The Function ρ(c) 12.4.3 p-Adic Case 12.4.3.1 Conventions 12.4.3.2 Equivalence Classes of Quasi-Characters 12.4.3.3 Choice of f 12.4.3.4 Fourier Transforms 12.4.3.5 The ζ-Functions 12.4.3.6 The Function ρ(c) 12.4.4 Non-vanishing of the Standard Local ζ Functions Problems References 13 The Global Theory 13.1 The Additive Group: The Adèles 13.1.1 The Field as a Subring of the Adèles 13.1.2 The Poisson Formula 13.1.3 Analogy with the Geometric Riemann-Roch Theorem 13.2 The Multiplicative Group: The Idèles 13.2.1 Multiplicative Fundamental Domain 13.2.2 The Quasi-Characters of Ik/k 13.3 Global Zeta Functions Problems Reference 14 Hecke L-functions 14.1 Characters of the Idèles Problems Reference 15 Recovering the Classical Theory 15.1 The Riemann Zeta Function 15.2 Dedekind ζ Functions 15.3 The General Case: L-functions of Characters 15.3.1 Idèlic Character 15.3.2 Choice of the Adèlic Function 15.3.3 Fourier Transform 15.3.4 The ζ-Function 15.3.5 Conclusion: Analytic Continuation and Functional Equation Problems 16 An Extended Example: The L-function of a CM Elliptic Curve 16.1 The L-function of an Elliptic Curve 16.2 Interpretation as a Hecke L-function 16.3 Interpretation as the L-function of a Representation Problems References Index