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L-FUNCTIONS Contents Introduction 1 1. Algebraic Number Theory 3 2 L-FUNCTIONS ALEKSANDER HORAWA Contents Introduction1 1. Algebraic Number Theory3 2. Hecke L-functions 10 3. Tate's Thesis: Fourier Analysis in Number Fields and Hecke's Zeta Functions 13 4. Class Field Theory 52 5. Artin L-functions 55 References 79 Introduction These notes, meant as an introduction to the theory of L-functions, form a report from the author's Undergraduate Research Opportunities project at Imperial College London under the supervision of Professor David Helm. The theory of L-functions provides a way to study arithmetic properties of integers (and, more generally, integers of number fields) by first, translating them to analytic properties of certain functions, and then, using the tools and methods of analysis, to study them in this setting. The first indication that the properties of primes could be studied analytically came from Euler, who noticed the factorization (for a real number s > 1): 1 X 1 Y 1 = : ns 1 − p−s n=1 p prime This was later formalized by Riemann, who defined the ζ-function by 1 X 1 ζ(s) = ns n=1 for a complex number s with Re(s) > 1 and proved that it admits an analytic continuation by means of a functional equation. Riemann established a connection between its zeroes and the distribution of prime numbers, and it was also proven that the Prime Number Theorem 1 2 ALEKSANDER HORAWA is equivalent to the fact that no zeroes of ζ lie on the line Re(s) = 1. Innocuous as it may seem, this function is very difficult to study (the Riemann Hypothesis, asserting that the 1 non-trivial zeroes of ζ lie on the line Re(s) = 2 , still remains one of the biggest open problems in mathematics). The ζ-function was extended further by Dirichlet, who proved that for coprime integers a; m 2 Z, there are infinitely many primes in the arithmetic progression (a + dm)d2Z. This theorem is clearly difficult to approach algebraically and Dirichlet's idea was to restate it in analytic terms. He considered a function that is defined just like ζ, but contains coefficients: for any function χ: Z ! C, the Dirichlet L-function is defined by: 1 X χ(n) L(s; χ) = : ns n=1 To ensure that the Euler factorization 1 X χ(n) Y 1 = ns χ(p) n=1 p 1 − ps holds, we assume that the function χ is multiplicative, i.e. χ(mn) = χ(m)χ(n) for (m; n) = 1. The way Dirichlet proved the theorem is by showing that Y L(s; χ); χ where χ varies over multiplicative functions with period m, i.e. χ(a + m) = χ(a), has a pole at s = 1. One can show that this statement implies that the series X 1 p p≡a mod m (where the sum is over primes p such that p ≡ 1 mod m) diverges, so there must be infinitely many primes in the progression (a + dm)d2Z. In fact, it gives a stronger result about the density of primes in the arithmetic progression (Dirichlet's Density Theorem). For the proofs of these theorems, see [Ser73, Chap. VI]. The aim of this exposition is to generalize Dirichlet L-functions further to encapture not only primes in Q, but also primes in finite algebraic extensions of Q, so-called number fields. In Section1, we provide the necessary background on algebraic number theory. The reader already familiar with [Ser79, Chap. I{III] or [Lan94, Chap. I{III] can skip ahead to Section2. In Sections2 and3, we define the analog of Dirichlet L-functions for an abelian extension| Hecke L-functions|and, following Tate's thesis [CF86, Chap. XV], use Fourier analysis in number fields to obtain a functional equation. In Sections4 and5, we use class field theory to define L-functions for non-abelian extensions|Artin L-functions|that agree with the Hecke L-functions in the abelian case, and use representation theory to obtain a function equation. Our presentation largely follows [Sny02]. Acknowledgements. These notes were prepared as part of a UROP project. I would like to thank my supervisor, Professor David Helm, for suggesting the topic, providing me with many explanations and resources, and hours of helpful discussions. I am also grateful to the Department of Mathematics at Imperial College London for the financial support. L-FUNCTIONS 3 1. Algebraic Number Theory We have seen that we can use the Dirichlet L-functions to study rational primes (primes of Z). We would like to generalize the ideas to study primes in other rings. We start by providing a background on algebraic number theory. Throughout most of this section, we will follow [Ser79, Chap. I{III] and [Lan94, Chap. I{III]|we will refer to particular sections where applicable. 1.1. Factoring Primes in Extensions. In this section, we follow [Ser79, Chap. Ix5] [Lan94, Chap. Ix5,7]. Suppose A is a Dedekind domain (a Noetherian integrally closed domain such that for every prime ideal p 6= 0 of A, Ap is a discrete valuation ring) and K is its field of fractions. Moreover, suppose L is a separable extension of K of degree n, and let B be the integral closure of A in L. Then B is also a Dedekind domain. We will denote by IK the set of fractional ideals of K. Recall that for any a 2 IK we have a unique factorization into primes Y a = pvp(a); p where we denote by vp the valuation at a prime ideal p of A (if a is an ideal of A, then vp(a) is the highest power of p dividing a, and we extend this to any a 2 IK ). Definition 1.1. If P is a non-zero prime ideal of B and p = P \ A, then we say that P divides p or that P lies above p, and we write Pjp. Definition 1.2. Suppose P divides p in the extension L=K. The ramification index eP of P over p in the extension L=K is the exponent of P in the factorization of p into prime ideals of B. In other words: Y p = PeP : Pjp The residue degree fP of P over p in the extension L=K is the degree of the extension of residues fields, B=P of A=p; symbolically fP = [B=P : A=p]: We may sometimes write eP=p and fP=p when the extension L=K is not implicitly clear. Proposition 1.3. Both the ramification index and the residue degree are multiplicative in towers: if Pjqjp in the tower of extensions L=M=K, then eP=p = eP=qeq=p; fP=p = fP=qfq=p: Proof. The proof is clear: one simply writes down the factorizations in the extensions for the first equality and the tower of extensions of residue fields for the second equality. 4 ALEKSANDER HORAWA Definition 1.4. When there is only one prime ideal P dividing p and fP = 1, then L=K is totally ramified at p. In that case p = PeP : When eP = 1 and B=P is separable over A=p, then L=K is unramified at P. If L=K is unramified at every P lying above p, then L=K is unramified at p. Finally, if eP = fP = 1, then p splits completely in L and there are exactly [L : K] primes of B above p. Proposition 1.5. Let p be a non-zero prime ideal of A. Then X [L : K] = ePfP: Pjp Proof. We may assume that A is a discrete valuation ring by localizing at p. Then B is a free A-module of rank n = [L : K], and B=p is an A=p-vector space of dimension n by [Lan94, Prop. 19, Chapter I]. Moreover, Y p = PeP ; Pjp and, since p ⊆ PeP , we have homomorphisms B ! B=p ! B=PeP which together yield Y ': B ! B=p ! B=PeP : Pjp Each B=PeP is an A=p-vector space, and hence so is the direct product. We have that Y ker ' = fb 2 B j b 2 PeP for each Pjpg = PeP = p: Pjp The Chinese reminder theorem shows that ' is surjective. Therefore Y B=p ∼= B=PeP Pjp as an A=p-vector space. To prove the proposition, it is enough to show that the dimension of eP B=P over A=p is ePfP. Let π be a generator of P in B and j ≥ 1 be an integer. Note that Pj=Pj+1 ⊆ Pj=pPj, since Pj+1 ⊇ pPj, so we can view Pj=Pj+1 as an A=p-vector space. In fact, the map B=P ! Pj=Pj+1 given by b 7! πjb is an A=p-isomorphism. Therefore, we have a decomposition series 1 ⊆ PeP−1=PeP ⊆ PeP−2=PeP ⊆ ::: ⊆ P=PeP ⊆ B=PeP with each quotient isomorphic to B=P. Hence: eP−1 eP X [B=P : A=p] = [B=P : A=P] = ePfP; i=0 as requested. L-FUNCTIONS 5 Definition 1.6. Let M be the Galois closure of L, and let G = Gal(M=K) and H = Gal(M=L), so that G=H can be identified with the embeddings L,! M which preserve K. We define the (relative) norm NL=K : IL ! IK to be the multiplicative function Y NL=K a = σa: σ2G=H Note that the norm sends a principal ideal αB to the principal ideal 0 1 Y @ σαA A; σ2G=H so can also define a map NL=K : L ! K by Y NL=K α = σα: σ2G=H 1.2.
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