LECTURE 2: ARTIN SYMBOL, ARTIN MAP, ARTIN RECIPROCITY LAW AND FINITENESS OF GENERALIZED IDEAL CLASS GROUP
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1 Clss Field Theory Study Seminr Jnury LECTURE 2: ARTIN SYMBOL, ARTIN MAP, ARTIN RECIPROCITY LAW AND FINITENESS OF GENERALIZED IDEAL CLASS GROUP YIFAN WU Plese send typos nd comments to Contents 1. Artin Symbol 1 2. Property of Artin Symbol 3 3. Cycles nd Artin Mp 4 4. Artin Reciprocity Lw 5 5. Admissible Cycles nd Kernel of Artin Mp 5 6. Generlized Idel Clss Groups 7 Reference: (1) VI 1, VII 4, X 1, Algebric Number Theory by Serge Lng (2) Notes of, Lecture 3, Mth 776 Winter 2011 by Krtik Prsnn (3) Algebric Number Theory by Jurgen Neukirch (4) Locl Fields by Jen-Pierre Serre Setting: 1. Artin Symbol L/K Glois extension of number fields with Gl(L/K) belin. p n unrmified prime idel in O K. P prime of O L bove p. In this setting, the inerti group E(P p) {σ Gl(L/K) : σ(α) α(mod P) α O L } 1 1
2 Clss Field Theory Study Seminr Jnury nd the decomposition group / D(P p) {σ Gl(L/K) : σ(p) P} Gl(O L /P O K /p) / The Glois group Gl(O L /P O K /p) is cyclic nd generted by Frobenius utomorphism x x #O K/p where #O K /p N(p), the norm the prime idel p. The Frobenius utomorphism lifts to n element φ(p p) D(P p) stifsying φ(p p)(α) x N(p) (mod P) α O K Recll the Glois group Gl(L/K) cts trnsitively on the set of ll primes of L lying bove p. If we consider nother prime σ(p) bove p then the decomposition group nd the inerti group re relted by D(σ(P) p) σd(p p)σ 1, E(σ(P) p) σe(p p)σ 1 nd the Frobenius element φ(σ(p) p) corresponding to σ(p) is relted by φ(σ(p) p) σφ(p p)σ 1 This is sying we get well-defined Frobenius conjugcy clss in Gl(L/K). Now our Glois group Gl(L/K) is belin, this conjugcy clss consists of only one element, i.e. φ(σ(p) p) φ(p p) σ Definition 1.1. The rtin symbol L/K : φ(p p) p Now consider the reltive discriminnt δ disc(l/k). This is n O K idel generted by ll elements of form det(t r L/K (β i β j )) with n tuples (β 1,, β n ) with entries β i O L, n [L : K]. Recll we hve Theorem 1.2. p rmifies if nd only if p disc(l/k). The Artin symbol mkes sense for those unrmified p, i.e. those p tht is prime of disc(l/k). Now let I(δ) :group of frctionl idels prime to δ. Definition 1.3. The mp L/K p p 2
3 Clss Field Theory Study Seminr Jnury extends to group homomorphism multiplictively I(δ) Gl(L/K), p v i i ri L/K L/K p i This is clled the Artin mp. 2. Property of Artin Symbol Proposition 2.1. (1) Let τ : L τl be n isomorphism tht does not necessrily fix K pointwisely,then τl/τk L/K τ τ 1 τ Note here L/K τl/τk Gl(L/K), Gl(τL/τK), Gl(τL/τK) τ Gl(L/K)τ 1 (2) If it hppens to be the cse tht τ fixes K pointwisely, τ Gl(L/K), then L/K L/K τk K, τl L, τ τ 1 τ (3) (Consitency Property) Let M/L/K be such tht M/K is belin, then the restriction to L stisfies M/K L/K L For this to mke sense, the primes occurring in the prime fctoriztion of shll not be rmified in M (we only know they re unrmified in L). (4) Let E/K be finite extension. Let q be prime of E lying bove p. Glois theory tells us LE/E is Glois nd Gl(LE/E) Gl(L/K) hence LE/E is belin, nd q is unrmified: L LE L E K If f f(q p) [O E /q : O K /p] is the inerti degree, then f LE/E L/K L q p 3 E
4 Clss Field Theory Study Seminr Jnury (5) Let E/K be finite extension. b be frctionl idel of E. Suppose for ll q occurring in the prime fctoriztion of b, nd q, we hve p q O K unrmified in L. Then LE/E L/K L b NK Eb (6) In prticulr if L/E/K then L/E b L/K NK Eb Proof. Not very hrd. See Serge Lng, pge Cycles nd Artin Mp Definition 3.1. A plce v of n lgebric number field K is n equivlence clss of non-trivil bsolute vlues of K. Denote v to be representtive of the plce v. The set of ll plces v is denoted s M K Recll tht in Mth 676 we ve seen tht: Definition 3.2. An bsolute vlue on field K is function : K R 0 stisfying (1) x 0 x 0 (2) xy x y (3) (Archimeden bsolute vlue) x + y x + y (4) (Non-Archimeden bsolute vlue) x + y mx{ x, y } Definition 3.3. The non-archimeden equivlence clsses re clled finite plces, nd denoted v. The Archimeden equivlence clsses re clled infinite plces, nd denoted v. From Mth 676, we ve lso seen tht Theorem 3.4. Those finite plces v re in one-to-one correspondence with the primes p of O K. Those infinite plces v comes from (1) either embeddings into R, τ : K R, in which the completion K v R; (2) or embeddings into C, τ : K C, in which the completion K v C. Definition 3.5. Let K be number field, cycle c of K is forml product c v M K v m(v) such tht n v 0 v, n v 0 for ll but finitely mny v m(v) is clled the multiplicity of v. We write v c if m(v) > 0. 4
5 Clss Field Theory Study Seminr Jnury Definition 3.6. We sy frction idel of K is prime to c if this frctionl idel is prime to p for ll those p tht corresponds to finite plce v M K with m(v) 0 (the correspondence is given by Theorem 3.4). We define I(c) : {frctionl idels of K prime to c} Let L/K be belin extension s before. As long s ll rmified primes of K divides c (more precisely, the plces corresponding these primes divide c), then sme s in Definition 1.3, we get Artin mp I(c) Gl(L/K), p v i i ri L/K L/K p i 4. Artin Reciprocity Lw There re two prts of Artin Reciprocity Lw: (1) To show for ll cycles c tht is divided by ll rmified primes of K, the Artin mp I(c) Gl(L/K) described bove is surjective. (2) To determine the kernel of the Artin mp. Prt (1) needs the so-clled norm index equlity. We will focus on Prt (2) now. 5. Admissible Cycles nd Kernel of Artin Mp We strt with short clcultion. Given ll the setting s bove. L/K belin, c v M K v m(v) cycle of K divisible by ll rmified primes. Define n(c) {N L/K (B) : B frctionl idels of L, v q (B) 0 for ll q p with m(p) > 0} This is subgroup of I(c) becuse ll such N L/K (B) is prime to c. We clculte: (L/L ) L/K B NL KB Id L Gl(L/K) where the left equlity is given by Proposition 2.1 (6) chnging E to L. This implies tht n(c) kernel of Artin mp. To identify the kernel precisely, we will need to define the notion of n dmissible cycle. To motivte this concept, we recll result from Mth 676: 5
6 Clss Field Theory Study Seminr Jnury Lemm 5.1. Suppose v is (finite or infinite) plce of K, w plce of L such tht w v nd w is unrmified over v. Complete K nd L s K v nd L w respectively. Let U Kv nd U Lw denote the units in O Kv nd O Lw respectively, then the norm function is surjective: N Lw/K v : U Lw U Kv Proof. From Mth 676 December 1, Definition 5.2. Let c v m(v) be cycle of K. Let v be plce of K, we define subgroup W c K v s: U Kv, the group of units, if v c W c (v) {x O Kv : x 1(mod p) m(v) }, if v is finite plce, p corresponds to v nd v c {x K v : x > 0}, if v c nd K c R Remrk 5.3. The convention is tht if v is infinite plce, then U Kv K v. We will lwys ssume tht the multiplicity m(v) 0 if K v C. If K v R it only mtters whether m(v) 0 or not. Definition 5.4. Let L/K be Glois extension. A cycle c of K is dmissible for L/K if W c (v) N Lw/K v (L w) for ll v nd w plce of L nd w v. The lst ingredient for identifying the kernel of Artin mp is Definition 5.5. We defined subgroup P c P (P is the group of principl frctionl idels) of K s: P c {(α) : α K, { m(v) α 1(mod p v ) if v finite plce nd v c σ v (α) > 0 if v is rel embedding nd v c } Note P c is subgroup of I(c), so P c n(c) is subgroup of I(c). Theorem 5.6. (Prt (2) of Artin Reciprocity Lw from Section 4) Let L/K be n belin extension. Let c be ny dmissible cycle in K, then the kernel of the Artin mp ( ) ker I(c) Gl(L/K) P c n(c) Remrk 5.7. There exist smllest dmissible cycle f for L/K mening f c for ll other dmissible cycles c, clled the conductor of L/K. 6
7 Clss Field Theory Study Seminr Jnury There exists commuttive digrm: I(f)/P f n(f) I(c)/P c n(c) Gl(L/K) In prticulr, I(c)/P c n(c) I(f)/P f n(f) for ny dmissible cycles. f δ, i.e., the discriminnt δ of L/K is lwys dmissible. 6. Generlized Idel Clss Groups We will study the group I(c)/P c before we eventully prove Artin Reciprocity Lw. Definition 6.1. Let K be number field, c be ny cycle of K, the group I(c)/P c is clled the group of c idel clsses. Remrk 6.2. These re viewed s generlized idel clss groups since when c 1, we just recover the idel clss groups of O K we ve seen before in Mth 676. Theorem 6.3. (Finiteness of Generlized Idel Clss Groups) I(c)/P c is finite. Proof. I ll do it on blckbord. The proof is bsed on Serge Lng, pge 125,
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