1 Algebraic number theory 笔记整理 Jachin Chen [email protected] 0 目 录 第 1 章 Valuation Theory 4 1.1 Valuation and valuation ring........................................................... 4 1.2 Discret valuations.............................................................................. 12 1.3 Complete the valuation field ........................................................... 19 1.4 Absolute values and Berkovich spaces .......................................... 21 1.5 Extension of valuations .................................................................... 27 1.6 Hensel’s Lemma................................................................................ 28 第 2 章 Dedekind domain 34 2.1 *Fractional ideals and ideal class groups....................................... 34 2.2 Etale´ algebras and lattices................................................................ 36 2.3 Prime decomposition in an order ................................................... 46 2.4 Norm map of one-dimensional Schemes....................................... 52 2.5 Ramification in Dedekind domain ................................................. 59 2.6 *Different and discriminant............................................................. 66 第 3 章 Ramification theory 73 3.1 Extensions of complete DVR ........................................................... 73 3.2 Higher Ramification Groups ........................................................... 80 3.3 *The Theory of Witt Vectors ............................................................ 88 3.4 Structure of complete DVRs ............................................................ 98 3.5 Lubin-Tate formal groups ................................................................ 107 第 4 章 Local and Global 115 4.1 Topological ring and group ............................................................. 115 4.2 Local fields and global fields........................................................... 120 4.3 Adele ring and Idele group ............................................................. 130 4.4 Ideal Class Groups and Unit Groups ............................................. 140 4.5 Some computations and applications ............................................ 144 目 录 3 第 5 章 Number Fields 145 5.1 *Cyclotomic fields ............................................................................. 145 5.2 Dirichlet character and Gauss sum................................................. 150 5.3 The Minkowski bound ..................................................................... 157 5.4 Dirichlet Unit Theorem .................................................................... 166 5.5 Moduli of a number field................................................................. 173 5.6 Some examples and supplements................................................... 181 第 6 章 Zeta and L-functions 186 6.1 Prime number theorem .................................................................... 186 6.2 The functional equation ................................................................... 193 6.3 *Dirichlet L-functions....................................................................... 200 6.4 The analytic class number formula ................................................ 205 6.5 Dirichlet density and Polar density................................................ 210 第 7 章 Class field theory 217 7.1 *Ray class groups and ray class fields............................................ 217 7.2 *Class field theory: ideal-theoretic form........................................ 224 7.3 *Local class field theory ................................................................... 227 7.4 *Global class field theory ................................................................. 233 7.5 Tate Cohomology .............................................................................. 240 1 第 1 章 Valuation Theory 1.1 Valuation and valuation ring Krull valuations Let G be an Abelian group. A subset ∆ j G is said to be an ordering of the group G if: (1) 8s1; s2 2 ∆; s1 + s2 2 ∆; (2) 8s 2 ∆; s 2 ∆ or −s 2 ∆; (3) ∆ \ −∆ = f0g For the given ordering ∆ of the group G we denote: a 6 b :() b − a 2 ∆ It is easy to check that the relation 6 is a linear ordering on G. We call G an ordered group. 注记. 1. ∆ = fa 2 G j a > 0g is an ordering. 2. Ordered groups do not have elements of finite order. For the ordered group G we define a projective group G [ f1g consistent of the group G with its ordering and a symbol 1 which satisfies the following conditions: 1. 8a 2 G; a < 1; 2. 8a 2 G; a + 1 = 1 + a = 1. 1.1. VALUATION AND VALUATION RING 5 ♦ 定义 1.1.1. Krull valuation Let F be a field and G [ f1g an ordered projective group. A surjective map v : F ! G [ f1g is said to be the Krull valuation when: 1. v(a) = 1 () a = 0; 2. v(ab) = v(a) + v(b); 3. v(a + b) > minfv(a); v(b)g, provided a + b =6 0. 注记. 1. v(1) = 0 and v(a−1) = −v(a); 2. If an = 1, then v(a) = 0; v(−1) = 0. 3. v(a) = v(−a). 引理 1.1.1. If v(a) =6 v(b), v(a + b) = minfv(a); v(b)g. Proof. We may assume that v(a) < v(b). Suppose that v(a + b) =6 minfv(a); v(b)g. This implies v(a + b) > minfv(a); v(b)g, in particular v(a + b) > v(a). Thus: v(a) = v((a + b)−b) > minfv(a + b); v(b)g > v(a) which is a contradiction. Valuation rings Let F be a field. A ring A ⊆ F is called the valuation ring if: 8a 2 F; a 2 A or a−1 2 A. If the field F is not given we shall assume that F is the field of fractions of A. ♦ 定理 1.1.1 Let v : F ! G [ f1g be a valuation. 1. The set Av = fa 2 F j v(a) > 0g is a valuation ring. We shall call it the valuation ring associated with v. 2. The set mv = fa 2 F j v(a) > 0g is the only maximal ideal in the ring Av. In particular, Av is a local ring and κ(v) := Av/mv is a field, which shall be called the residue field of v. 3. The set Uv = fa 2 F j v(a) = 0g is a group consistent of all units of the ring Av. Proof. Easy check. 6 CHAPTER 1. VALUATION THEORY 注记. It is easy to see that R is a valuation ring iff the divisibility relation j is total. ajb :() ba−1 2 R ♦ 定理 1.1.2 Let A be a valuation ring in F . There exists a Krull valuation v : F ! G [ f1g s:t: × Av = A; mv = AnA Proof. Consider the quotient additive group G := F ×/A×. Define the relation by a + A× 6 b + A× :() ajb such that the group G is an ordered abelian group. Define the mapping v : F ! G [ f1g by 8 < × a + A if a =6 0 v(a) := : 1 if a = 0 We shall show that v is a valuation which will be called the canonical valua- tion. Fix a; b 2 F . We may assume that v(a) 6 v(b). Note that v(1 + ba−1) > 0, therefore − v(a + b) = v(a) + v(1 + ba 1) > v(a) = minfv(a); v(b)g If v1 : F ! G1 [ f1g and v2 : F ! G2 [ f1g are two valuations, then we say that they are equivalent, written v1 ' v2, if there exists an order preserving group isomorphism g : G1 ! G2 s:t: v2 = g ◦ v1 (we take g(1) = 1). Clearly such relation is an equivalence and we can state the following result: The set of all equivalence classes of the relation ' is in a bijective corre- spondence with the family of all valuation rings in F . Proof. Suppose that v1 : F ! G1 [ f1g and v2 : F ! G2 [ f1g are equivalent. > > Then v1(a) 0 iff v2(a) 0, so Av1 = Av2 . Conversely, let A be a valuation ring and let A = Av for some valuation v : ! [ f1g F G . By the previous theorem A = AvA for the cannonical valuation 1.1. VALUATION AND VALUATION RING 7 × vA. We shall show that v ' vA. Observe that vjF × : F ! G is a surjective × homomorphism and that ker vjF × = A . By the isomorphism theorem GA := × × F /A ' G. If g : GA ! G is such isomorphism, then it is easy to verify that g preserves order and that v = g ◦ vA. Rank of valuations Let G be an ordered abelian group. A subgroup H of G is said to be the isolated subgroup if 8 h 2 H; fg 2 G j 0 6 g 6 hg ⊆ H The set G(G) of all isolated subgroups of G is totally ordered by inclusion. The order type of the set G(G)nfGg is called the rank of G. If G is a value group of some valuation v, then the rank of valuation v is the rank of G. () −1 () −1 2 × () Note that v(a) = v(b) v(ab ) = 0 ab Rv a; b associates −1 −1 in Rv, so v v(I) = I for ideal in Rv. On the other hand, vv (H) = H for any subset of G since v is surjective. Therefore we have established a bijection between Spec(Rv) and the set of isolated subgroups of G by p 7! v(Rvnp) −1 H 7! Rvnv (H+) ♦ 定义 1.1.2. Rank of valuation The rank of valuation v is defined by rank of G or dimension of Rv. 注记. Since Spec(Rv) is ordered by inclusion, dim(Rv) = jSpec(Rv)j − 1. Clearly v is a valuation of rank 0 iff G is the zero group. Moreover, we have 引理 1.1.2. v : F ! G [ f1g has rank less or equal that 1 iff G is Archimedean, that is: 8 a; b > 0; 9 n 2 N s:t: na > b S Proof. (Sketch) Suppose that G =6 0. Fix a 2 G and let Ha = fb 2 G j −na 6 n2N b 6 nag. Note that Ha is an isolated subgroup of G. Next we shall show that if H an isolated subgroup of G and a 2 H, then Ha ⊆ H. 8 CHAPTER 1. VALUATION THEORY Thus Ha is the smallest isolated subgroup containing a. This implies that the rank of G is equal to 1 iff: 8 a > 0;Ha = G which is equivalent to 8 a; b > 0; 9 n 2 N s:t: b 6 na. ♦ 命题 1.1.1. Holder Every ordered Archimedean commutative group is isomorphic to some subgroup of the (R; +). Proof. (Sketch) Fix 0 6 g 2 G. Define the mapping Φ: G ! R by: n o m Φ(h) := inf j mg > nh; m; n 2 Z n Since it is obvious that Φ is order-preserving, so we need to check that 1. Φ is well-defined; 2. Φ is a homomorphism; 3. Φ is injective. The valuation whose value group is a subgroup of (R; +) is called the expo- nential valuation.