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Class Field Theory Franz Lemmermeyer Class Field Theory April 30, 2007 Franz Lemmermeyer email [email protected] http://www.rzuser.uni-heidelberg.de/~hb3/ Preface Class field theory has a reputation of being an extremely beautiful part of number theory and an extremely difficult subject at the same time. For some- one with a good background in local fields, Galois cohomology and profinite groups there exist accounts of class field theory that reach the summit (exis- tence theorems and Artin reciprocity) quite quickly; in fact Neukirch’s books show that it is nowadays possible to cover the main theorems of class field theory in a single semester. Students who have just finished a standard course on algebraic number theory, however, rarely have the necessary familiarity with the more advanced tools of the trade. They are looking for sources that include motivational material, routine exercises, problems, and applications. These notes aim at serving this audience. I have chosen the classical ap- proach to class field theory for the following reasons: 1. Zeta functions and L-series are an important tool not only in algebraic number theory, but also in algebraic geometry. 2. The analytic proof of the first inequality is very simple once you know that the Dedekind zeta function has a pole of order 1 at s = 1. 3. The algebraic techniques involved in the classical proof of the second inequality give us results for free that have to be derived from class field theory in the idelic approach; among the is the ambiguous class number formula, Hilbert’s Theorem 94, or Furtw¨angler’s principal genus theorem. 4. Many of the central unsolved problems in modern number theory are directly connected to analytic objects. Let me just mention the Riemann conjecture for various L-functions, the Stark conjectures, the conjecture of Birch and Swinnerton-Dyer, and the whole Langlands program. I also have tried to approach certain central results by first treating special cases; this is not particularly elegant, but it helps students to see how some of the more technical proofs evolved from relatively simple considerations. vi Table of Contents Part I. Dirichlet’s Analytic Methods 1. Dirichlet Series for Quadratic Characters ................. 3 1.1 Euler . 3 1.2 Basic Properties of the Riemann Zeta Function . 7 1.3 Quadratic Number Fields . 12 1.4 Gauss . 13 1.5 Dirichlet’s L-series . 15 2. The Nonvanishing of L(1, χ) for Quadratic Characters .... 21 2.1 Dirichlet’s Proof for Prime Discriminants . 21 2.2 Nonvanishing of Dirichlet’s L-functions . 27 2.3 Computation of L(1, χ) ................................. 29 3. Primes in Arithmetic Progression......................... 41 3.1 Characters. 42 3.2 Primes in Arithmetic Progression . 45 3.3 Cyclotomic Number Fields . 47 4. Dirichlet .................................................. 53 4.1 Dirichlet’s L-series for Quadratic Forms . 53 4.2 Genus Theory for Quadratic Number Fields. 55 4.3 Primes with Prescribed Residue Characters . 56 4.4 Primes Represented by Binary Quadratic Forms . 58 5. Algebraic Number Fields ................................. 61 5.1 Archimedean Valuations of a Number Field . 61 5.2 Arithmetic of Number Fields . 62 5.3 Prime Decomposition in Relative Extensions . 65 5.4 Prime Ideals in Galois Extensions . 67 5.5 Minkowski Bounds. 70 6. Dirichlet’s Unit Theorem ................................. 77 6.1 Units in Quadratic Number Fields . 77 6.2 Dirichlet’s Unit Theorem . 79 viii Table of Contents 6.3 The Unit Theorems of Minkowski and Herbrand . 82 7. Dedekind’s Zeta Function ................................. 87 7.1 Distribution of Ideals . 88 7.2 Dirichlet’s Class Number Formula . 95 7.3 Cyclotomic Fields . 96 8. Density Theorems ........................................ 97 8.1 Kronecker’s Density Theorem . 97 8.2 Frobenius Density Theorem for Abelian Extensions . 100 8.3 Kummer Extensions . 102 8.4 Decomposition Laws in Kummer Extensions . 105 8.5 Density Theorems of Kummer and Hilbert . 106 Part II. Hilbert Class Fields 9. The Hilbert Class Field ................................... 111 9.1 Weber’s Motivation√ . 111 9.2 The Field Q(√−5) ..................................... 114 9.3 The Field Q( 3)....................................... 115 9.4 Hilbert Class Field Theory II . 116 10. The First Inequality ...................................... 123 10.1 Weber’s Inequality . 123 10.2 Proof of the First Inequality . 126 10.3 Consequences of the First Inequality . 128 11. The Second Inequality .................................... 133 11.1 Preliminaries. 133 11.2 The Second Inequality for Unramified Extensions . 136 11.3 The Ambiguous Class Number Formula . 138 11.4 The Herbrand Quotient of the Unit Group . 142 12. Examples of Hilbert Class Fields ......................... 147 13. The Artin Symbol ........................................ 149 13.1 Inertia Groups . 149 13.2 The Symbols of Frobenius and Artin . 150 13.3 The Artin Isomorphism . 153 14. Frobenius Density ........................................ 157 14.1 Frobenius and his Density Theorem . 157 14.2 Group Theoretical Preliminaries . 162 14.3 Prime Ideal Decomposition in Nonnormal Extensions . 163 14.4 The Proof of Frobenius’ Density Theorem . 165 Table of Contents ix Part III. Takagi’s Class Field Theory 15. Ideal Groups ............................................. 171 15.1 Generalized Class Groups . 171 15.2 Takagi’s Class Field Theory . 175 15.3 The Fundamental Inequalities . 180 16. Artin’s Reciprocity Law .................................. 183 16.1 Cyclotomic Fields . 183 16.2 Base Change . 183 16.3 Proof of Artin’s Reciprocity Law . 183 17. The Existence Theorem .................................. 185 18. Norm Residues and Higher Ramification ................. 187 18.1 Higher Ramification Groups . 187 Part IV. Appendix A. Gamma, Theta, and Zeta ................................. 193 A.1 Euler’s Gamma Function . 193 A.2 Jacobi’s Theta Functions . 193 A.3 Riemann’s Zeta Function . 193 A.4 Quadratic Gauss Sums . 193 B. A Beginner’s Guide to Galois Cohomology................ 195 B.1 H1(G, A).............................................. 195 B.2 Hb 0(G, A).............................................. 195 B.3 Hb −1(G, A)............................................. 195 B.4 Galois Cohomology for Cyclic Groups . 195 B.5 Herbrand’s Lemma . 195 B.6 Capitulation . 195 B.7 Ambiguous Ideal Classes . 195 C. Solutions of Selected Problems ........................... 197 Bibliography .................................................. 203 x Table of Contents Part I Dirichlet’s Analytic Methods 1 1. Dirichlet Series for Quadratic Characters Analytic methods occupy a central place in algebraic number theory. In this chapter we introduce the basic tools of the trade provided by Dirichlet. Most of the results proved here will be generalized step by step in subsequent chapters until we finally will have all the techniques required for the proof of the First Inequality of class field theory. Most modern accounts of class field theory give an arithmetic proof of both the First and the Second Inequality. This approach has the additional advantage of bringing out clearly the local-global aspects of class field the- ory. On the other hand, class number formulas and the density theorems of Dirichlet, Kronecker, Frobenius and Chebotarev are central results of al- gebraic number theory which every serious student specializing in number theory must be familiar with, in particular since these analytic techniques are also needed in the theory of elliptic curves (or, more generally, abelian varieties) and modular forms. In this theory, the analog of the class number formula of Dirichlet and Dedekind is the conjecture of Birch and Swinnerton- Dyer, which – together with the Riemann hypothesis – belongs to the most important open problems in number theory. 1.1 Euler One of the earliest outstanding results of Euler was the formula π2 1 1 1 = 1 + + + + .... (1.1) 6 4 9 16 This is the value ζ(2) of Riemann’s zeta function 1 1 1 ζ(s) = 1 + + + + .... 2s 3s 4s Euler’s first “proof” of (1.1) was full of holes, but very beautiful. In a nutshell, here’s what he did. sin x Fix some α ∈ R with sin α 6= 0, and consider the function f(x) = 1− sin α . This function has a Taylor expansion 4 1. Dirichlet Series for Quadratic Characters x x3 x5 f(x) = 1 − + + − .... sin α 3! sin α 5! sin α The real roots of this function are x = 2nπ + α and x = (2n + 1)π − α. Euler knew that two polynomials of degree n with equal roots and equal constant term (the value at x = 0) must be the same. Regarding f(x) as a polynomial of infinite degree, he concluded that ∞ Y x x f(x) = 1 − 1 − 2nπ + α (2n + 1)π − α n=−∞ ∞ x Y x x = 1 − 1 − 1 + α (2n − 1)π − α (2n − 1)π + α n=1 x x 1 − 1 + . 2nπ + α 2nπ − α Expanding the right hand side and comparing coefficients yields ∞ 1 1 X 1 = + sin α α (2n − 1)π − α n=1 1 1 1 − + − , (1.2) (2n − 1)π + α 2nπ + α 2nπ − α ∞ 1 1 X 1 = + 2 α2 ((2n − 1)π − α)2 sin α n=1 1 1 1 − + − . (1.3) ((2n − 1)π + α)2 (2nπ + α)2 (2nπ − α)2 π Putting α = 2 in (1.2) gives Leibniz’s series π 1 1 1 = 1 − + − ± .... 4 3 5 7 π For α = 4 , (1.2) produces π 1 1 1 1 √ = 1 + − − + + ..., 2 2 3 5 7 9 which Euler credits to Newton; in fact, this formula appears in a letter from Newton to Oldenberg from October 24, 1676. π Plugging α = 2 into (1.3) gives 1 1 π2 1 + + + ... = . 32 52 8 Euler then observes that 1 1 1 ζ(2) = 1 + + + ... + ζ(2), (1.4) 32 52 4 1.1 Euler 5 π2 and this then implies ζ(2) = 6 . Euler’s arguments for the product expansion of f(x) are not convincing for two reasons: first, he only considered real roots of f; second, the functions f(x) and exf(x) have the same roots and the same constant term, so these properties do not determine f.
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