ELEMENTARY IWASAWA THEORY FOR CYCLOTOMIC FIELDS
HARUZO HIDA
In this topic course, assuming basic knowledge of algebraic number theory and com- mutative algebra, we pick topics from the theory of cyclotomic fields. Our treatment is elementary. We plan to discuss the following four topics: (1) Class number formulas, (2) Basics of cyclotomic fields and Iwasawa theory, (3) Stickelberger’s theorem, (4) Cyclicity over the Iwasawa algebra of the cyclotomic Iwasawa module. Since this is a topic course, for some of the topics, we just give the results without detailed proofs. Main reference is Chapters 4, 5, 6, 7 and 10 of the following book [ICF]: [ICF] L. C. Washington, Introduction to Cyclotomic fields, Graduate Text in Mathe- matics 83, 1997. Here is a relevant book: [LFE] H. Hida, Elementary Theory of L–functions and Eisenstein Series, LMSST 26, Cambridge University Press, Cambridge, 1993. Here is an Overview of the goal of the lectures. We write Q for the field of all algebraic numbers in C. Any finite extension of Q inside Q is called a number field. Write µN for the group of N-th roots of unity inside Q× and Q(µN ) for the field generated by roots of unity in µN , which is called a cyclotomic field. For any given number field K, the class group ClK defined by the quotient of the group of fractional ideals of K modulo principal ideals is a basic invariant of K. Itisa desire of many algebraic number theorists to know the module structure of ClK. Or if K/Q is a Galois extension, GK/Q := Gal(K/Q) acts on ClK. Thus it might be easier to see the module structure of ClK over the group ring Z[GK/Q] larger than Z. The first step towards this goal of determining ClK for K = Q[µN ] was given in 1839 by Dirichlet as a formula of the order of the class group (his class number formula). The cyclotomic field K has its maximal real subfield K+ and K/K+ is a quadratic extension if N is odd with GK/K+ generated by complex conjugation c. The norm map gives rise + to a homomorphism Cl Cl := Cl + whose kernel is written as Cl− (the minus K → K K K part of ClK). By the formula, if N is an odd prime p, the order of the ClK− is given by p 1 − 1 1 2p ( χ− (a)a) (Dirichlet/Kummer). × −p χ:(Z/pZ)× Q ;χ( 1)= 1 j=1 →Y − − X
Date: April 14, 2018. The author is partially supported by the NSF grant: DMS 1464106. 1 ELEMENTARY IWASAWA THEORY 2
a Since G = (Z/pZ)× sending σ G with σ (ζ) = ζ (ζ µ ) to a (Z/pZ)×, K/Q ∼ a ∈ K/Q a ∈ p ∈ we have Z[GK/Q] ∼= Z[(Z/pZ)×]. Since each character χ of GK/Q extends to an algebra homomorphism χ : Z[GK/Q] Q sending σa to χ(a), Kummer–Stickelberger guessed that → p 1 p 1 − − a 1 1 1 θ := σ− annihilates Cl− as χ(θ )= ( χ− (a)a). 0 p a K 0 p a=1 j=1 X X This “symbolic” statement means that Aβθ0 (for any fractional ideal A of K) is principal as long as βθ0 Z[GK/Q] for β Z[GK/Q]. Writing a for the Z[GK/Q]-ideal generated by elements of∈ the form βθ Z[∈G ], we might expect: 0 ∈ K/Q a ClK− ∼= Z[GK/Q]/ ? (Cyclicity over Z[GK/Q]) which is not generally true. After supplying basics of cyclotomic fields, we will prove in the course Stickelberger’s theorem:
Cl− Z = Z [G ]−/(a Z )− (p-Cyclicity) K ⊗Z p ∼ p K/Q ⊗ p assuming Kummer–Vandiever conjecture: p - Cl+ (see [BH] for numerical examples). | K| Here A− = x A cx = x for complex conjugation c for an ideal A of Z [G ]. Set { ∈ | − } p K/Q Λ= Zp[[T ]] (one variable power series ring). Then we can easily prove that
lim Zp[GQ[µ n ]/Q] = Λ[µp 1] (lim σ1+p t =1+ T ), p ∼ − 7→ ←−n ←−n where the limit is taken via restriction maps GQ[µ ]/Q σ σ Q[µ n] GQ[µ n ]/Q. pn+1 3 7→ | p ∈ p Then, assuming again Kummer–Vandiever conjecture, we further go on to show Iwa- sawa’s way of proving his main conjecture and cyclicity of his Iwasawa module X := lim (Cl− Zp): n Q[µpn] ⊗ ←− X = Λ[µp 1]−/(Lp) ∼ − for the T -expansion Lp of the Kubota–Leopoldt p-adic L-function.
Contents 1. Cyclotomic fields 3 2. An outline of class field theory 4 3. Relative class number formula and Stickelberger’s theorem 6 4. Basic properties of Gauss sum 7 5. Prime factorization of Gauss sum 10 6. A consequence of the Kummer–Vandiver conjecture 14 7. Kummer theory 16 8. Cyclicity theorem for F0 = Q[µp] 17 9. Proof of the cyclicity theorem 18 10. Iwasawa theory 20 11. An asymptotic formula of An− 21 12. Cyclotomic units | | 24 + 13. Class number formula for Fn 25 References 29 ELEMENTARY IWASAWA THEORY 3
1. Cyclotomic fields We recall basic structure theory of cyclotomic fields. We write Q for the field of all algebraic numbers in C. Any finite extension of Q inside Q is called a number field. In algebraic number theory, the theory of cyclotomic fields occupies very peculiar place. It was the origin of the development of algebraic number theory and still inspires us with many miraculous special features. 2πi n Fix a prime p > 2 and consider a primitive root of unity ζp := exp( pn ). Then ζp Xp 1 2 p 1 p 1 j − satisfies the equation Φ1(X)= X −1 = 1+X+X + +X − = j=1(X ζp). Note that (X+1)p 1 p p − j 1 p 1 ···p 2 − Φ (X +1) = − = X − = X − + pX − + + p which is an Eisenstein 1 X j=1 j ··· Q polynomial. Therefore Φ1(X + 1) is irreducible, and hence Φ1(X) is irreducible. In the n P Xp 1 pn−1 same manner, ζ n is a root of Φ (X) = − = Φ (X ) is irreducible. Therefore p n Xpn−1 1 1 − n 1 Q(ζ n )/Q is a field extension of degree equal to deg(Φ ) = p − (p 1). In particular, p n − in Q(ζpn )/Q, p fully ramifies with ζpn 1 giving the unique prime ideal (ζpn 1) over pn−1(p 1) − − p; so, (ζ n 1) − = (p) in Q[ζ n ], and Z [ζ n 1] is the p-adic integer ring of p − p p p − Q [ζ n ]. For all other prime l = p, taking a prime l of Q[ζ n] above l, ζ n := (ζ n mod l) p p 6 p p p n j is a primitive p -th root in a finite field of characteristic l; so, ζpn are all distinct for n 1 j = 1,...,p − (p 1). Thus Q [ζ n ] is unramified at l, and Z [ζ n ] is an unramified − l p l p valuation ring over Zl; so, it is the l-adic integer ring of Ql[ζpn ]. This shows (cf. [CRT, 9]) §
Lemma 1.1. The ring Z[ζpn ] is the integer ring of Q[ζpn] and the roots of Φn(X) (i.e., n all primitive p -th roots gives rise to a basis of Z[ζpn ] over Z, p fully ramifies in Z[ζpn ], and ζ n 1 generates a unique prime ideal of Z[ζ n ] over p. p − p
Since all Galois conjugate of ζpn is again a root of Φn(X), Q[ζpn] is a Galois extension. σ νn(σ) n n Since σ GQ[ζpn ]/Q is determined by ζpn = ζp for νn(σ) (Z/p Z)×, we have a ∈ n ∈ character G n (Z/p Z)× which is injective. Since the LHS and RHS have the Q[ζp ]/Q → same order, νn is an isomorphism. Writing the residue field of l - p as F = Z[ζpn ]/l, we l have F = Fl[ζpn ]. Then the Frobenius element Frobl GF/Fl = Dl sends ζpn to ζpn . Thus we get ∈
Lemma 1.2. For a prime l = p, we have ν (Frob )=(l mod pn). 6 n l
It is customary to write µ n Q× for the cyclic group generated by ζ n . Since p ⊂ p Q[ζpn ] contains µpn , we write hereafter Q[µpn ] for Q[ζpn ] freeing the notation from a choice of a generator of µpn . Write Q[µp∞ ] := n Q[µpn ]. Then the restriction map Resm,n(σ)= σ Q[µ n ] for σ GQ[µ m ]/Q (m > n) gives the following commutative diagram: | p ∈ p S νm m G m (Z/p Z)× Q[µp ]/Q −−−→ Res mod pn νn n GQ[µpn ]/Q (Z/pZ)×. y −−−→ y ELEMENTARY IWASAWA THEORY 4
Passing to the limit, we get ν (1.1) GQ[µp∞ ]/Q Zp× with ν(Frobl)= l for l = p. −→∼ 6 p Let Γ:= 1+ pZp Zp×. Then Zp×/Γ ∼= (Z/pZ)× which has order p 1. Thus z z ⊂ − pn pn−17→ is the identity map on (Z/pZ)×. On the other hand, for any z Zp×, z − 1 n n n n 1 pn pn−1 1∈ ≡ mod p as (Z/p Z)× has order p p − . Thus z z p pn . We therefore have pn − | − p | ≤ a limit ω(z) = limn z which satisfies plainly ω(z) = ω(z); i.e, ω(z) µp 1 and →∞ ∈ − ω(z) z mod p. Thus µp 1 Zp× is a cyclic subgroup of order p 1. We thus have ≡ − ⊂ − Zp× = Γ µp 1. Thus we get × −
Lemma 1.3. Let Dl be the decomposition subgroup of GQ[µp∞ ]/Q of a prime l. Then if l = p, D is the infinite cyclic subgroup topologically generated by Frob isomorphic to 6 l l l Zp ω(l)Z. If l = p, D is equal to the inertia subgroup I which is the entire group h i × p p ∞ GQ[µp∞ ]/Q. In particular, for each prime, the number of prime ideals in Zp[µp ] over l is finite and is equal to [GQ[µp∞ ]/Q : Dl].
e(l) Remark 1.4. More generally, for an integer N, making prime factorization N = l l , e(l) e(l) e(l) e(l) 1 Q[µ ] is the composite of Q[µ e(l)] with degree (Z/l /Z)× = ϕ(l )= l l − . In N l | | − Q the field Q[µle(l) ], the prime l ramifies fully and all other primes are unramified. Therefore the fields Q[µle(l)] l N are linearly disjoint over Q, and hence { } | e(l) GQ[µ ]/Q = GQ[µ ]/Q = (Z/l Z)× = (Z/NZ)×. N ∼ le(l) ∼ ∼ l N l N Y| Y| This also tells us that the integer ring of Q[µN ] is given by Z[µN ] ∼= l Z[µle(l)]. The decomposition subgroup of a prime q - N in GQ[µN ]/Q is isomorphic to the subgroup N e(l) of (Z/NZ)× generated by the class (q mod N). If l N, Il is isomorphic to (Z/l Z)× (l) e(l) | and writing N = N/l , Dl is isomorphic to the product of Il and the subgroup of (l) (Z/N Z)× generated by the class of l. Since the Galois group GQ[µN ]/Q is generated by inertia groups of primes l N, for any intermediate extension Q E F Q[µN ], some prime ramifies. In other words,| if F/E is unramified everywhere,⊂ we⊂ conclude⊂ E = F .
Exercise 1.5. e(l) Give a detailed proof of the fact: Z[µN ] ∼= l Z[µl ] and Z[µN ] is the integer ring of Q[µN ]. N Exercise 1.6. Prove that for a finite set S = l ,...,l of primes. prove that the { 1 r} number of prime ideals over any given prime l in Z[µ ∞ ,...,µ ∞ ] is finite. l1 lr
2. An outline of class field theory We give a brief outline of class field theory (a more detailed reference is, for example, [CFT] or [BNT]), which will be covered in Math 205c in Spring 2018. For a given number field F with integer ring O, pick an O-ideal n, let In be the set of all fractional n ideals prime to n. We put On := lim O/n (the n-adic completion of O). Write the n e(l) ←− prime factorization of n = l n l . Then by Chinese reminder theorem, we have On = | ∼ Q ELEMENTARY IWASAWA THEORY 5
e(l) O/l Ol. We write O for Ol F (which is the localization l (l) ∩ β Q O = α, β O with αO + l = O . (l) α ∈
Exercise 2.1. Prove the above identity. e(l) For a principal ideal (α) In, we write α 1(mod n)× if α 1+ l Ol for all prime ∈ ≡ ∈ l n. More generally, we write α β(mod n)× for (α), (β) In if α/β 1(mod n)×. | ≡ ∈ ≡ e(l) Exercise 2.2. Prove that α 1(mod n)× if and only if α 1+ l O(l) for all prime l n. ≡ ∈ | Define
Pn := (α) In α 1(mod n)× { ∈ | ≡ } (2.1) + P := (α) Pn σ(α) > 0 for all field embeddings σ : F R . n { ∈ | → } + If F has no real embedding (i.e., F is totally imaginary, we have Pn = Pn . + Exercise 2.3. Let F = Q[√5]. Is PO = PO true? How about Q[√15]? + + Then Cln := In/Pn (resp. Cln = In/Pn ) are called the (resp. strict) ray class group modulo n of F . They are finite groups. We have written ClF for ClO. The order ClO is called the class number of F . Here is a main theorem of class field theory: | |
Theorem 2.4. For each n as above, there is a unique abelian extension Hn/F (resp. + Hn /F ) such that + (1) Prime ideals prime to n is unramified in Hn/F and Hn /F , and every real embed- ding of F extends to a real embedding of Hn; in particular, if all Galois conjugates of F are in R (i.e., F is totally real), Hn is totally real; + (2) Cln = GH /F and Cl = G + by an isomorphism sending the class of prime ∼ n n ∼ Hn /F + ideal l prime to n in Cln (resp. Cln ) to the corresponding Frobl GHn/F (resp. ∈ + G + ), where Frob is a unique element in G for K = H , H such that Hn /F l K/F n n N(l) Frobl(l)= l and Frobl(x) x mod l for N(l) := O/l ; ≡ | | + (3) For any finite abelian extension K/F , there exists an O-ideal n such that K Hn (the ideal maximal among n with K H+ is called the “conductor” of K);⊂ ⊂ n (4) For a finite extension F/F0 with integer ring O0, we write Hn/F and Hn0 /F 0 (resp. + + Cln and Cl0n for the corresponding class fields (resp. the corresponding class groups). Then we have H0 Hn and the following commutative diagram n ⊃ Res G 0+ 0 G + H n /F −−−→ Hn /F
o o
N 0 x+ F /F x+ Cl0 Cl . n −−−−→ n Here NF 0/F is induced by the norm map sending a O0-prime ideal L prime to n f σ 0 to l (l = L O) with f =[O0/L : O/l] (note NF /F L = σ G 0 L O if F 0/F ∩ ∈ F /F ∩ is a Galois extension). Q ELEMENTARY IWASAWA THEORY 6
Example 2.1. Suppose F = Q. Consider the map n Z nZ + n = Z n (n) + { ∈ | } 3 7→ ∈ Cl(N) for each integer n > 0 prime to N. This induces an injective homomorphism + (Z/NZ)× ClN . For each (α) IN , take an integer n prime to N with α n(mod N)× → ∈ + ≡ and α/n > 0. Then the class of (α) and (n) in ClN coincide by definition. Therefore, + + we get ClN ∼= (Z/NZ)×, and hence HN = Q[µN ]. Exercise 2.5. What is H H+ = Q[µ ]? Determine G as a quotient of N ⊂ N N HN /Q (Z/NZ)×.
Corollary 2.6 (Kronecker–Weber–Hilbert). Any finite abelian extension of Q is con- tained in Q[µN ] for some positive integer N.
3. Relative class number formula and Stickelberger’s theorem
Let Fn := Q[µpn+1 ]. Complex conjugation c acts non-trivially on Fn as νn(c) = 1. + + pn(p 1) − − Let Fn be the fixed field of c; so, [Fn : Q]= 2 . We have the following field diagram
, F → H n −−−→ n
∪ ∪ x+ , x+ Fn → Hn −−−→ + + for the Hilbert class fields Hn/Fn and Hn /Fn . Write the class group of Fn (resp. + + + + F ) as Cl and Cl . Define then Cl := Ker(N + : Cl Cl ). Since H is n n n n− Fn/Fn n n n + + → real, H Fn = F ; so, G + = G + G + + . Thus the restriction map n n Hn Fn/Fn ∼ Fn/Fn Hn /Fn ∩ × + Res : GH /F G + + is onto. Therefore N + : Cln Cl is onto. Recall n n → Hn /Fn Fn/Fn → n Dirichlet’s L-function of a character χ : (Z/NZ)× C× defined by an absolutely and locally uniformly converging sum for s C with Re(→s) > 1: ∈ ∞ s L(s,χ)= χ(n)n− , n=1 X where we use the convention that χ(n)=0if n and N are not co-prime. The function L(s,χ) extends to a meromorphic function on the whole complex plane C with only possible pole at s = 1, and if χ is non-trivial, it is holomorphic everywhere. Here is a formula of Cl− (e.g. [ICF, Theorem 4.17]): | n | Theorem 3.1. Let m = n + 1. Then we have
m 1 1 Cl− = 2p L(0,χ− ) | n | 2 χ:(Z/pmZ)× C×;χ( 1)= 1 →Y − − 1 1 pf 1 f with L(0,χ− ) = χ− (a)a, where χ is a primitive character modulo p with − pf a=1 0 < f m. ≤ P ELEMENTARY IWASAWA THEORY 7
By functional equation for primitive character modulo N (see [ICF, Chapter 4] or [LFE, 2.3]): § τ(χ)(2π/N)sL(1 s,χ−1) − 2Γ(s)cos(πs/2) if χ( 1) = 1, L(s,χ)= τ(χ)(2π/N)sL(1 s,χ−1) − − ( 2√ 1Γ(s) sin(πs/2) if χ( 1) = 1 − − − N 1 with the Gauss sum τ(χ)= a=1 χ(a) exp(2π√ 1a/N), L(0,χ− ) is almost L(1,χ)/2πi − 1 which is directly related to the class number. Because of this, we put χ− in the formula, though we can replace themP by χ for an obvious reason.
We would like to study the group structure of Cln = ClFn and the module structure of m Cln over Z[GFn/Q] = Z[(Z/p Z)×](m = n+1). We first describe Kummer-Stickelberger ∼ n theory to determine the annihilator of Cln− in Z[(Z/p Z)×]. Writing σa GFn/Q for the σa a m ∈ element with ζ = ζ (ζ µpm ) for a (Z/p Z)×, from the above class number ∈ ∈ pm a 1 m formula, Kummer/Stickelberger guessed that θ = θn := a=1 pm σa− Z[(Z/p Z)×] 1 ∈ would kill Cln as χ(θn) + L(0,χ− ). More generally, let F/Q be an abelian extension P a 1 with F Q[µN ] for a minimal N > 0. Define θF := a (Z/NZ)× N σa− F Q[G] for ⊂ ∈ | ∈ G = GF/Q, where 0 x < 1 is the fractional part of a real number x (i.e., x x Z) ≤{ } Pa −{ } ∈ and σ G sends every N-th root of unity ζ to ζ for a (Z/NZ)×. a ∈ Q[µN ]/Q ∈ Theorem 3.2 (Stickelberger). Pick β Z[G] such that βθ Z[G]. Then for any ∈ F ∈ fractional ideal a of F , aβθF is principal.
4. Basic properties of Gauss sum The idea of proving this to compute prime factorization of the Gauss sum G(χ) of a finite field F of characteristic p and to show roughly (G(χ)) = aθF . Write Tr : F F → p for the trace map. Then for any character χ : F× Q×, the Gauss sum is defined to be → 2πi G(χ) := χ(a)ζTr(a) for ζ := exp( ), − p p p a F X∈ where we put χ(0) = 0 as before. Note that G(χ) = τ(χ) if χ is a character of − 2πi Tr(a) F× = (Z/pZ)×. The character ψ : F C× given by ψ(a) = exp( ) is non-trivial p → p as F is generated by primitive N-th root of unity for N = ( F 1) whose minimal N N 1 | |− polynomial is a factor of X + X − + + 1 (i.e., Tr(a) = 0 for some a F). ··· 6 ∈ Exercise 4.1. Give a detailed proof of the fact: Tr(a) = 0 for some a F; so, Tr : F 6 ∈ → Fp is onto. Lemma 4.2. We have Tr(ab) (1) a F χ(a)ζp = χ(b)G(χ) for b F×, − ∈ ∈ (2) G(χ)= χ( 1)G(χ), P (3) If χ = 1 for− the identity character 1, G(χ)G(χ)= χ( 1) F , (4) If χ =6 1, G(χ) 2 = F . − | | 6 | | | | Since τ(χ)= G(χ) if F = F , we get the corresponding formula for τ(χ). − p ELEMENTARY IWASAWA THEORY 8
Proof. The assertion (1) holds by the variable change ab a combined with χ(b) = 1 7→ χ− (b). Then we have
Tr(a) Tr(a( 1)) (1) G(χ)= χ(a)ζ− = χ(a)ζ − = χ( 1)G(χ) − p − p − a F a F X∈ X∈ proving (2). Note that for c = 1, 6 Tr(b(c 1)) ζ − = 1 p − b F× X∈ Tr(b(c 1)) as 1+ b F× ζp − = 0 (character sum). We then have ∈ P 1 Tr(a b) G(χ)G(χ)= χ(ab− )ζp − a,b F× X∈ −1 c=ab Tr(bc b) Tr(b(c 1)) = χ(c)ζp − = χ(1) + χ(c) ζp − b,c F× b F× c=0,1 b F× X∈ X∈ X6 X∈ =( F 1) + χ(c)( 1) = F . | |− − | | c=0,1 X6 This finishes the proof (3), and (4) follows from (2) and (3).
For two characters ϕ,φ : F× Q×, we define the Jacobi sum as → J(ϕ,φ) := ϕ(a)φ(1 a). − − a F X∈ See [We1] and [We2] for amazing properties of Gauss sum and Jacob sum which was the origin of Weil’s conjecture (Riemann hypothesis for zeta function of algebraic varieties over finite fields), which was solved by P. Deligne. Here are some such properties: Lemma 4.3. (1) J(1, 1) = 2 F , (2) J(1,χ)= J(χ, 1) = 1 if χ−|= 1,| (3) J(χ, χ)= χ( 1) if χ = 1, 6 − 6 (4) J(ϕ,φ)= G(ϕ)G(φ) if ϕ = 1,φ = 1,ϕφ = 1. G(ϕφ) 6 6 6 Proof. Since there are F 2 elements in F 0, 1 , we get (1). When one character is 1 and the other not, the| |− Jacobi sum is the sum−{ over} F 0, 1 , and hence the result (2) follows. −{ } To show (4), we set ϕ(0) = φ(0) = 1 and we compute G(ϕ)G(φ):
Tr(a+b) G(ϕ)G(φ)= ϕ(a)φ(b)ζp a,b F X∈ a+b c =7→ ϕ(a)φ(c a)ζTr(c) = ϕ(a)φ(c a)ζTr(c) + ϕ(a)φ( a). − p − p − a,c F a F,c F× a F X∈ ∈X∈ X∈ If ϕφ = 1, we have 6 ϕ(a)φ( a)= φ( 1) ϕφ(a) = 0. − − a F a F× X∈ X∈ ELEMENTARY IWASAWA THEORY 9
As for the first sum, without assuming ϕφ = 1, we have 6 ϕ(a)φ(c a)ζTr(c) a=bc ϕ(c)φ(c)ϕ(b)φ(1 b)ζTr(c) = G(ϕφ)J(ϕ,φ). − p − p a F,c F× b F,c F× ∈X∈ ∈X∈ Therefore we get G(ϕ)G(φ)= G(ϕφ)J(ϕ,φ) as desired if ϕφ = 1. Suppose now ϕφ6 = 1. Then we have ϕ(a)φ( a)= φ( 1) 1(a)= φ( 1)( F 1). − − − | |− a F a F X∈ X∈ Thus 1 ϕ( 1) F = G(ϕ)G(ϕ− )= ϕ( 1)( F 1) + G(1)J(ϕ, ϕ) − | | − | |− Tr(a) Tr(a) with G(1)= a F× ζp = ( a F ζp 1) = 1. This shows (3). − ∈ − ∈ − Corollary 4.4.PSuppose that ϕNP= φN = 1 for 0 < N Z. Then G(ϕ)G(φ) is an ∈ G(ϕφ) algebraic integer in Q(µN ). Let 0 < N Z, and suppose p - N. Then Q(µ ) and Q(µ ) is linearly disjoint as p is ∈ N p unramified in Q(µN ) while p fully ramify in Q(µp). Thus
G = G G = (Z/NZ)× (Z/pZ)×. Q(µpN )/Q ∼ Q(µN )/Q × Q(µp)/Q ∼ × Let σ G be the automorphism of Q(µ ) corresponding to (a, 1) (Z/NZ)× a ∈ Q(µpN )/Q pN ∈ × (Z/pZ)× for a (Z/NZ)×. ∈ a N G(χ) a σa Lemma 4.5. If χ = 1 and a is an integer prime to Np, then σ = G(χ) − G(χ) a ∈ Q(µ ) and G(χ)N Q(µ ). N ∈ N Proof. Since ζσa = ζa for ζ µ and ζσa = ζ , we have ∈ N p p G(χ)σa =( χ(x)ζTr(x))σa = χa(x)ζTr(x) = G(χa). − p − p x F x F X∈ X∈ σ b Similarly, for σ GQ[µpN ]/Q[µN ], we have some 0 < b Z prime to p such that ζp = ζp. Then we have ∈ ∈ σ Tr(x) σ Tr(bx) bx b 1 G(χ) =( χ(x)ζ ) = χ(x)ζ =7→ χ− (b)G(χ). − p − p x F x F X∈ X∈ Replacing χ by χa, we get a σ a a G(χ ) = χ− (b)G(χ ). a G(χ) a σa Thus σ fixes , and hence we get G(χ) − Q(µ ). Taking a := 1+ N, we get G(χ)σa ∈ N the last assertion. Here is the last lemma in this section: Lemma 4.6. We have G(χp)= G(χ). ELEMENTARY IWASAWA THEORY 10
p p Proof. The Frobenius automorphism Frobp of F acts Frobp(a) = a . Thus Tr(a ) = Tr(Frobp(a)) = Tr(a). Then we have
p G(χp)= χp(a)ζTr(a) = χ(ap)ζTr(a ) = χ(Frob (a))ζTr(Frobp(a)) − p − p − p p a F a F a F X∈ X∈ X∈ Frobp(a) a = 7→ χ(a)ζTr(a) = G(χ). − p a F X∈ as desired.
5. Prime factorization of Gauss sum
Let I0 = I0 := β Z[G] βθ Z[G] for G = G for an abelian extension F/Q. F { ∈ | F ∈ } F/Q We put sF := I0θF =(θF ) Z[G] which is called the Stickerberger ideal of F . We start with the following lemma.∩
Lemma 5.1. Suppose F = Q[µN ] and put G := GF/Q. Then the ideal I0 is generated by c σ for c Z prime to N. − c ∈ Proof. We first show that I0 I00 := (c σ ) in Z[G]. We have plainly ⊃ − c c a ac 1 (c σ )θ = c σ− Z[G] − c N − N a ∈ a X n o n o which shows the result. Now we show the converse. Suppose that x = x σ I0. Then a a a ∈ −1 a 1 ac b,cPa ab 1 xθ = x σ− 7→= 7→ x σ− . c N ac−1 a N b c a ! a ! X X n o Xb X 1 The coefficient of σ1− is given by a x a x a x a a a mod 1. a N ≡ N ≡ N a a X n o X n o P Thus x 0 mod N. Since N = (1+ N) σ I00, we find a a ≡ − 1+N ∈ P xaσa = xa(σa a)+ xa I00. − ∈ a a a X X X Thus I0 I00. ⊂ Theorem 5.2 (Stickelberger). Let F = Q[µN ]. Then sF annihilates ClF ; i.e., for any βθ β I0 and a fractional ideal A of F , A for θ := θ is principal. ∈ F x xaσa Here for x = a xaσa, A = a A . We prepare several lemmas before going into the final phase of the proof. The idea is to make primeP factorizatioinQ of Gauss sums which gives a canonical generator of pβθ for a prime p. Let p be a prime and F be a finite extension of F ; so, F = pf =: q. Let p be a p | | prime ideal of Z[µq 1] above p. Since F× is a cyclic group of order q 1, we have an − − ELEMENTARY IWASAWA THEORY 11
isomorphism ω = ωp : F× = µq 1 Z[µq 1]×. Since all q 1-th roots of unity are ∼ − ⊂ − − distinct modulo p, we may assume that ω(a) mod p = a F×. ∈ Step 1: We analyze the exponent of P appearing in the Gauss sum. Pick a prime vL(A) P p in Z[µ(q 1)p]. Write prime factorization of an ideal A of L L . If A = (α) is | − i principal, we simply write vL(α) for vL(A). Simply write v(i) for vP(G(ω− )). Here is a lemma on the behavior of the exponent v. Q Lemma 5.3. (1) v(0) = 0; (2) 0 v(i + j) v(i)+ v(j); (3) v(≤i + j) v(≤i)+ v(j) mod (p 1); (4) v(pi)= v≡(i); − q 2 f (5) i=1− v(i)=(q 2)f(p 1)/2 if q = p ; (6) v(i) > 0 if i 0− mod(−q 1); P 6≡ − 1 2 (7) v(1) = 1, in particular, G(ω− ) π mod P . ≡ Tr(a) G(ϕ)G(φ) Proof. Since G(1) = 1 as a F ζp = 0 (character sum), we get (1). Since G(ϕφ) is an ∈ algebraic integer (Corollary 4.4), (2) follows. Moreover, again by Corollary 4.4, G(ϕ)G(φ) P G(ϕφ) is an algebraic integer in the smaller field Q[µN ] in which p does not ramify, the difference v(i + j) (v(i)+ v(j)) is divisible by the ramification index p 1 of P/p. Therefore, we get (3).− By the existence of Frobenius automorphism on F,− we have G(χp) = G(χ) i i i i f (Lemma 4.6), which shows (4). Since G(ω− )G(ω )= G(ω )G(ω ) = q = p as long as ωi = 1 (i.e., 1 i pf 2), v( i)+ v(i) = f(p 1) as p 1 is± the ramification± 6 ≤ ≤ − − − − index of P/(p), summing these up, we get (5). To show (6), we put π := ζp 1 which is a generator of the unique prime in Z[µ ] (Lemma 1.1); so, π P. Then we− see p ∈ i i Tr(a) i G(ω− )= ω− (a)ζ ω− (a) 0 mod P, − p ≡− ≡ a a X X 1 2 which shows v(i) > 0. Now we prove (7) by showing G(ω− ) π mod P . Bya computation similar to the case of (6), we see ≡
1 1 Tr(a) 1 Tr(a) G(ω− )= ω− (a)ζ = ω− (a)(1 + π) − p − a a X X 1 2 1 2 ω− (a)(1 + πTr(a)) mod P π ω− (a)Tr(a) mod P . ≡− ≡− a a X X 2 f 1 Note that GF/Fp = Frobp = 1, Frobp, Frobp,..., Frobp− . Thus we have Tr(a) = f−1 h i { } a + ap + + ap . Therefore, we get ··· 1 1 p pf−1 ω− (a)Tr(a) a− (a + a + + a ) mod p. ≡ ··· a 0=a Z[µ ]/p X 6 ∈XN pb 1 pb 1 Since a a − is a non-trivial character of F× if 0
Step 2: Prime factorization of G(χ): To clarify the notation, fix a positive integer N, and choose a prime p - N. Write f for the order of the class of p in (Z/NZ)×. Thus f d N p 1 (exactly). Let χ := ω− for d =(q 1)/N; so, χ has values in µN and therefore N| − − χ = 1. Let R be a complete representative set for (Z/NZ)×/ p . Let p be a prime h i over p of Z[µq 1] such that ω(a) mod p = a F×, and put p0 = p Z[µN ] (which is the − ∈ ∩ −1 base prime for the prime factorization of G(χ)). Then by Remark 1.4, pσa a R is { 0 | ∈ } the set of all distinct primes above (p) in Z[µN ]. Let P0 be the unique prime above p0 in Z[µ ] as any prime above p fully ramifies in Z[µ ]/Z[µ ]. Let (resp. ) be a pN pN N P0 P0 prime in Z[µq 1] (resp. Z[µ(q 1)p]) over p0 (resp. P0). − − Lemma 5.4. Let the notation be as above. Then we have e −1 −1 Pa∈R v(ad)σa σa v(ad) (G(χ)) = P0 = P0 . a R Y∈ −1 −1 −1 σa σa σa Proof. Note that P0 is the unique prime above p0 . Write P := P0 . Then we have σa a a vP(G(χ)) = vP0(G(χ) )= vP0 (G(χ )) = v (G(χ )) = v(ad). Pe0 −1 σa This shows that the exponent of P = P0 in G(χ) is given by v(ad). Step 3: Determination of v(i) via p-adic expansion:
Lemma 5.5. Let 0 i f 1 Proof. Expand a = a0(a)+ a1(a)p + + af 1(a)p − . Then we have ··· − j j j+1 j+f 1 p a = a0(a)p + a1(a)p + + af 1(a)p − . ··· − f j+k Since p 1 mod(q 1), once the exponent j + k of p in ak(a)p exceeds f, we can remove p≡f modulo q −1. Thus we have − f 1 j 1 − − j k ` p a ak j (a)p + af+` j (a)p mod (q 1). ≡ − − − Xk=j X`=0 k ` f 1 Here p , p runs through 1,p,...,p − once for each, and aj(a) for j = 0, 1 ...,f 1 appears once for each. { } − Since the right-hand-side is less than q 1, we have − j f 1 k j 1 ` p a − ak j(a)p + − af+` j (a)p mod (q 1) = k=j − `=0 − − . q 1 q 1 − P P − Now we sum up over j. By moving j from 0 to f 1, modulo q 1, for each pk, the term a (a)pk shows up once for each i with 0 i − f 1. Then− each term involving i ≤ ≤ − ak(a) is given by f f 1 p 1 q 1 a (a)(1 + p + + p − )= a (a) − = a (a) − . k ··· k p 1 k p 1 − − Thus we conclude f 1 − pj a 1 q 1 1 v(a) = − a (a)= a (a)= . q 1 q 1 p 1 k p 1 k p 1 j=0 X − − − Xk − Xk − q 1 Then we see for d = N− , f 1 f 1 − pj ad − pj a v(ad)=(p 1) =(p 1) − q 1 − N j=0 j=0 X − X as desired. N NθF Corollary 5.7. We have (G(χ) )= p0 in Z[µN ]. τ τ a Proof. Since G(χ) = χ(a)G(χ) for GQ[µpN ]/F with ζp = ζp (Lemma 4.2 (1)), we have N G(χ) Z[µN ]. Since Q[µpN ]/F fully ramifies at p0 with ramification index p 1, we (p∈ 1) − find P0 − = p0. We have f 1 − j 1 p a 1 v(ad)σ− =(p 1) σ− . a − N a a (Z/NZ)×/ p j=0 a (Z/NZ)×/ p ∈ X h i X ∈ X h i σp Note that p0 = p0 as σp is the generator of the decomposition group of p; so, the effect of σapj on p0 is the same as the effect of σa on p0. Thus we conclude j j f−1 p a −1 f−1 p a −1 a −1 P P × σa Pj=0 Pa∈(Z/NZ)×/hpi N σ j ( ) P × σa j=0 a∈(Z/NZ) /hpi{ N } { } ap ∗ a∈(Z/NZ) { N } p0 = p0 = p0 . ELEMENTARY IWASAWA THEORY 14 The last equality ( ) follows from the bijection (Z/NZ)× p (Z/NZ)×/ p Then by Lemma 5.6, ∗ ↔ h i× h i N N(p 1)θF NθF (G(χ) )= P0 − = p0 as desired. Step 4: Proof of Stickelberger’s theorem: Let A be an ideal of Q[µN ] prime to (N). Write the prime factorization of A as i pi (here pi0s may overlap). Let χi be the d q 1 character of (Z[µ ]/p )× = F× given by ω for d = − for q = N(p ) = F . By N i Q pi N i | | Corollary 5.7 applied to each pi, we have NθF NθF N A = pi =( G(χi) ). i i Y Y Write γ := G(χ ) Q[µ ] for P = p . If βθ Z[G] (G = G ), then i i ∈ P N i i F ∈ F/Q Nβθ βN Q A QF =(γ ). β b σ β Since γ F by Proposition 5.1 and G(χ) − b F (Lemma 4.5), we have F [γ ]/F is a Kummer∈ extension adding N-th root γβ of γNβ∈ F . Now we claim that ∈ (Ur) F (γβ)/F can ramify only at prime factors of N. Here is the proof of (Ur): By adding N-th root, only ramified primes in the extension are factors of N and prime factors of γNβ. Since (γNβ) is N-th power of ideal A, for a βN β N prime factor l of γ , Fl[γ ] = Fl[ √u] for a unit u of the l-adic completion Fl. Since N l - N, Fl[ √u]/Fl is unramified. By definition, F F [γβ] Q[µ ]= F [µ ]. ⊂ ⊂ NP P The only primes ramifying in F [µP ]/F is factors of P which is prime to N. Therefore by (Ur), F [γβ] is unramified everywhere over F and is abelian over Q, which is impossible by Remark 1.4 unless F = F [γβ]. Therefore, Aβθ =(γβ) with γβ F . ∈ 6. A consequence of the Kummer–Vandiver conjecture + + Recall F = Q[µ n+1 ] with m = n + 1. Let h = Cl + = Cl . The following n p n Fn n conjecture is well known but we do not have a theoretical| | evei|dence| except for the conjecture numerically verified to be true valid for primes up to 163 million [BH]: Conjecture 6.1 (Kummer–Vandiver). p - h+ (so, Cl Z = 0). 0 0 ⊗Z p Suppose hereafter Conjecture 6.1. Let A± := Cl± Z (the p-Sylow part of Cl±, and n n ⊗Z p n put Rn := Zp[GFn/Q]. Then An± is a module over Rn. Define sn := Rn θFn Rn. Since R = Z[G ] Z , we have s = s Z , and hence by Stickelberger’s∩ theorem n Fn/Q ⊗Z p n Fn ⊗Z p (Theorem 3.2), we have snAn− = 0 without assuming Conjecture 6.1. Here Iwasawa’s cyclicity theorem which is a goal of this course: Theorem 6.2 . + (Iwasawa) Suppose p - h0 . Then we have an isomorphism An− ∼= Rn−/sn− as R -modules, where X− = (1 c)X (the “ ” eigenspace of complex conjugation c). n − − We start preparing several facts necessary for the proof of the theorem. We first compute the index [Rn− : sn−]. ELEMENTARY IWASAWA THEORY 15 Lemma 6.3. We have [R− : s−]= A− . n n | n | Proof. Let G = GFn /Q and θ = θFn . Let Tr := σ G σ Rn. Since z + z = 1 and ∈ ∈ { } {− } cσa = σ a, (1+ c)θ is equal to − P a 1 a 1 a a 1 σa− + σ−a = + − σa− = Tr. pm pm − pm pm a (Z/pmZ)× a (Z/pmZ)× a (Z/pmZ)× ∈ X ∈ X ∈ X 1 c 1+c Tr 1 c Let θ± := ± θ = θ θ = θ − R = R−. Thus we have 2 − 2 − 2 ∈ 2 n n 1+ c 1 (6.1) θ+ := θ = Tr. 2 2 1 c 1 c Since s = R θR , we have s− = − R − θR = R− θ−R . n n ∩ n n 2 n ∩ 2 n n ∩ n We first compute the index [Rnθ− : sn−]=[Rnθ− : Rn− θ−Rn]. Take x = b (Z/pmZ)× xbσb + 1 ∩ ∈ ∈ R . Since θ = Tr R , we have xθ− R− xθ R . Then n 2 ∈ n ∈ n ⇔ ∈ n P −1 1 1 1 a b b 1 xθ = ax σ− σ = ax σ −1 =7→ ax σ . pm b a b pm b ba pm ab b a a a Xb X Xb X Xb X m This shows xθ− R− xθ R ax 0 mod p for all b prime to p. But ∈ n ⇔ ∈ n ⇔ a ab ≡ 1 1 m axab b− (Pab)xab b− axa mod p . ≡ ≡ a a a X X X Therefore m xθ− R− xθ R ax 0 mod p . ∈ n ⇔ ∈ n ⇔ a ≡ a X m Thus Rnθ− Rn− = xθ− Rn a axa 0 mod p . Then Rnθ−/(Rnθ− Rn−) , m ∩ { ∈ | ≡m } ∩ → Z/p Z by sending xθ− to s xaa mod p . This map is surjective taking x := σ1 as P m s xaa = 1. Therefore we conclude [Rnθ− : Rnθ− Rn−]= p . P ∩ m Consider the linear map T : R− R− given by the multiplication by p θ−. Then P n → n m 1 [R− : p R−θ]= det(T ) − n n | |p m[F +:Q] 1 m G /2 m m m = p n L(0,χ) − = p | | − A− =[R−θ− : p R−θ−]p− A− . | |p | n | n n | n | χ Y by the class number formula (Theorem 3.1). Thus m [Rn− : p Rn−θ][Rnθ− : Rnθ− Rn−] − An = m ∩ . | | [Rn−θ− : p Rn−θ−] m m Since R− s− p R−θ and R−θ s− p R−θ, we have n ⊃ n ⊃ n n ⊃ n ⊃ n m [Rn− : p Rn−θ] [Rn− : sn−] m = . [Rn−θ− : p Rn−θ−] [Rn−θ− : sn−] From this we see [Rn− : sn−]][Rnθ− : Rnθ− Rn−] [Rn− : sn−][Rn−θ− : sn−] An− = ∩ = =[Rn− : sn−]. | | [Rn−θ− : sn−] [Rn−θ− : sn−] This shows the desired index formula. ELEMENTARY IWASAWA THEORY 16 7. Kummer theory p Let F/F0 be a finite extension inside Q. If α F × is not a p-power in F ×, F [√α]/F is a Galois extension of degree p, as a complete set∈ of conjugates of √p α over F is given by p p a p ζ√α ζ µp . Thus GF [ √p α]/F ∼= Z/pZ by sending σa GF [ √p α]/F with σa(√α)= ζp √α { | ∈ } p ∈ p to a Aut(µp)= Z/pZ. The extension F [√α]/F only depends on α mod (F ×) . Thus ∈ p p we simply write F [√α] for the extension corresponding to α F ×/(F ×) . For a subset p p p ∈ B of F ×/(F ×) , we put F [√B] := F [√α]α B, which is a (p, p, . . . , p)-Galois extension of F . ∈ Conversely, we take a p-cyclic extension K/F with a fixed isomorphism GK/F ∼= Z/pZ with σ G corresponding to 1 Z/pZ. By the normal basis theorem in Galois ∈ K/F ∈ theory, K is free of rank 1 over the group algebra F [GK/F ]. Thus for any character ξ : Z/pZ µp, the ξ-eigenspace F [ξ] = x K σ(x) = ξ(σ)x for all σ GK/F is one dimensional→ over F . Suppose that ξ(σ{) =∈ζa.| Pick β F [ξ]. then α∈:= βp } F a p ∈ ∈ as it is invariant under G . Since K F [√p α] and F [√p α] has degree p over F , we K/F ⊃ conclude K = F [√p α]. Thus every (p, p, . . . , p)-extension of F is of the form F [√p B]. p a p Since F [√α ] F [√α] for any a Fp, replacing B by the span of B in F × Z Fp, we ⊂ ∈ p ⊗ may assume that B is an F -vector subspace of F × F = F ×/(F ×) . p ⊗Z p Lemma 7.1. Let the notation be as above. Suppose dimFp B < . We have a non- ∞p p degenerate pairing , : G p B µ given by τ,β := τ(√β)/√β. h· ·i F [ √B]/F × → p h i p p Proof. If τ,β = 1 for all τ G p , plainly √β F ×; so, β = 0 in F ×/(F ×) . h i ∈ F [ √B]/F ∈ Thus the pairing is non-degenerate on the B side. Thus dim G p dim B. Fp F [ √B]/F ≥ Fp p p a p p p p Since F [√β] F [√β ] for a Fp and F [√αβ] F [√α, √β], we find that F [√B]= ⊃ ∈ ⊂ p F [√p β ,..., √p β ] for a basis β ,...,β of B over F ; so, pdimFp B = pd [F [√B]: F ]= 1 d { 1 d} p ≥ G p , which shows dim G p = d, and this finishes the proof. | F [ √α]/F | Fp F [ √B]/F Lemma 7.2. Suppose F F0 = Q[µp]. Let K/F be the maximal (p, p, . . . , p)-extension ⊃ 1 √p unramified outside p. Then for B := O[ p ]× Z Fp, we have K = F [ B] and dimFp B < . ⊗ ∞ Proof. Let L := F [√p α]. By multiplying p-power of a non-zero integer, we may assume that α O 0 . Write prime factorization of (α) = le(l). Plainly if l - p and ∈ −{ } l p - e(l), l ramifies in L/F . As for p, residually, any p-th root does not give any non- Q trivial extension as Frobp just raises p-power. Thus p can ramify independent of e(l) 1 for l p. This shows the result. By Dirichlet’s unit theorem, rank O[ ]× [F : Q]+ | Z p ≤ the number of prime factors of p = r. Thus dim B r. Fp ≤ Exercise 7.3. Compute dimFp B in Lemma 7.2. Corollary 7.4. Let the notation be as in Lemma 7.2. Assume F = Fn. Then K = p F [√B] for B generated by O× and 1 ζ m (m = n + 1). − p This is because the unique prime ideal p over p in Fn is principal generated by 1 ζpn+1 by Lemma 1.1. − ELEMENTARY IWASAWA THEORY 17 8. Cyclicity theorem for F0 = Q[µp] Let F = Fn and L = Ln/Fn be the maximal p-elementary abelian extension unramified everywhere. Here elementary means that Gal(Ln/Fn) is killed by p. Write An for the maximal p-abelian quotient Cl Z of Cl . Since L/F is elementary p-abelian, n ⊗Z p n Gal(L/F ) is an Fp-vector space. By class field theory (and Galois theory), Gal(L/F ) = p ∼ Cln/pCln = An/pAn. By Kummer theory, L = F [√B] for an Fp-vector subspace B p p of F × Z Fp = F ×/(F ×) . Since F [√b]/F is everywhere unramified for b F × with ⊗ p p ∈ b = (b mod (F ×) ) B, the principal ideal (b) is a p-power a for an O-ideal a. The ∈ p p p p class of a in An only depends on the class of b modulo (F ×) as F [√a b]= F [√b]. Thus p sending b := b mod (F ×) to the class of a, we get a homomorphism φ : B A [p] = → n x An px = 0 , which is obviously Zp[G]-linear for G := GF/Q. Assume b Ker(φ). { ∈ | p} p p p ∈ Then (b)=(a) ; so, b = a ε with ε O×. In other words, F [√b] = F [√ε]. Thus we p ∈ conclude that Ker(φ) O×/(O×) = O× F as Z [G]-modules. ⊂ ⊗Z p p Now we assume that F = F0. Then G = Hom(G, Zp×) is generated by Teichm¨uller character ω of order p 1. Then Zp[G] = i mod p 1 Zpei for the idempotent ei = 1 i − − ω− (σ)σ. For a finite p-abelian groupb H, we define p-rank(H) = dim H F = G σ L Fp ⊗Z p dim| | H[p] for H[p] = x H px = 0 (which is the minimal number of generators of PFp H by the fundamental{ theorem∈ | of finite} abelian groups); thus, H is cyclic if and only if p-rank(H) = 1. We first prove Theorem 8.1. Let A be the p-Sylow subgroup of Cl0 and put A = i eiA. If i is even and j is odd with i + j 1 mod(p 1), then we have ≡ − L p-rank(e A) p-rank(e A) 1+ p-rank(e A). i ≤ j ≤ i This implies a famous result of Kummer (when he proved FLT for regular primes): + p Cl p Cl− . | 0 |⇒ | 0 | Proof. By class field theory, A/pA Gal(L/F ) for the maximal p-abelian elementary ex- ∼= tension unramified everywhere. Then we have perfect Kummer pairing as in Lemma 7.1 i , : A/pA B µp. Note that σaa = ω (a)a for all a (Z/pZ)× if a Ai := eiA. i+k h· ·i ω(a×) → σa σa σa i k ∈ ω (a) ∈ Since a,b = a,b = a ,b = ω (a)a, ω (a)b = a,b for b Bk = ekB, a,b =h 1i unless ih+ ki 1h mod(p i 1).h This shows thati h , iindices a perfect∈ pairing hon Ai B . Thus dim≡A = dim −B . h· ·i i × j Fp i Fp j Now φ : B A [p]= e (A[p]) is Z [G]-linear. Since Ker(φ) B e (O× F ) and j → i i p ∩ j ⊂ j ⊗Z p Fp if k is even, k 0 mod(p 1); or k 1 mod(p 1), Ek := ek(O× Z Fp) ∼= 6≡ − ≡ − ⊗ (0 otherwise by Dirichlet’s unit theorem, we have (8.1) p-rank(A ) = dim B dim E + dim A [p] if l + k 1 mod(p 1). l k ≤ k k ≡ − If i is even (and j is odd), taking k = i (and l = j), we have p-rank(A ) = dim B dim E + dim A [p]=1+ p-rank(A ). j i ≤ i i i ELEMENTARY IWASAWA THEORY 18 If j is odd and j 1 mod(p 1), we have 6≡ − p-rank(A ) dim A [p]= p-rank(A ). i ≤ j j Thus if j 1 mod(p 1), the result follows. Suppose6≡ that j 1− mod(p 1). Then i = 0. Let L be the subfield of L with ≡ − 0 GL0/F = A0. From the exact sequence: 1 G G G 1 → L0/F → L0/Q → → with G acting on the normal subgroup GL0/F by conjugation, GL0/Q is abelian. Then I for the inertia subgroup I at p in GL0/Q is isomorphic to GF/Q; so, L0 is everywhere unramified extension of Q; so, A0 = G I is trivial. ∼ L0/Q Consider c =1+ p. Then ω(c σc)=1+ p 1 = p. For the Stickelberger element 1 p 1 1 − 1 − θ = − σ− a, ω((c σ )θ ) = aω− (a) 1(p 1) mod p kills A ; so, A = 0. p a=1 a − c F a ≡ − 1 1 This shows the result when j 1 mod(p 1). P ≡ P − Corollary 8.2. Assume the Kummer–Vandiver conjecture. Then we have A0− ∼= R0−/s0− as R0-modules for R0 := Zp[GF0/Q]. Proof. By Kummer–Vandiver, p-rank(eiA) = 0 for i even. Then by the above theorem (Theorem 8.1), p-rank(Aj) 1; so, Aj is cyclic. Thus we have a surjective module ≤ homomorphism ejR0 Aj. We write the image of ej in Aj as ej. A0− = j:odd ejA = R e = R ( e ); so, A− is cyclic. Since s kills A− and A− =[R− : s−] by j:odd 0 j 0 j:odd j 0 0 0 | 0 | L 0 0 Lemma 6.3, we conclude A− = R−/s−. P P 0 ∼ 0 0 9. Proof of the cyclicity theorem We first recall Nakayama’s lemma: Let R be a local ring with a unique maximal ideal mR and M be finitely generated R-module. Lemma 9.1 (NAK). If M = mRM, then M = 0. See [CRT, p.8] for a proof of this lemma This can be used as follows to determine the number of generators: Take a basis m1,..., mr of M/mRM over R/mR and lift it to m M so that (m mod m )= m . j ∈ j R j Corollary 9.2. The elements m1,...,mj generate M over R, and r is the minimal number of generators. Proof. For the R-linear map π : Ar M given by (a ,...,ar) a m , Coker(π) → 1 7→ j j j ⊗R R/mR = 0 as mj generate M R R/mR. Thus Coker(π) = 0 by NAK; so, m1,...,mr generate M. ⊗ P We recall the cyclicity theorem: + Theorem 9.3 (Iwasawa). Suppose p - h0 . Then we have an isomorphism An− = Rn−/sn− + ∼ as Rn-modules and An = 0. ELEMENTARY IWASAWA THEORY 19 p 1 Proof. We have already proven the result when n = 0. Note that R0 = i=0− eiR0 and eiR0 = Zp as a ring. Note that the restriction map GFn0 /Q σ σ Fn GFn/Q induces ∼ n 3 7→ | ∈Q a surjective ring homomorphism πn0 : Rn0 Rn for n0 > n. Take i Rn projecting 2 j →2 0 1 ∈ down to ei. Since ei = ei (so, ei = ei), i i. Since (πn)− (eiR0) is a p-profinite pj ≡ ring, limj j converges to an idempotent, lifting ej. We again wrote this lift as ej; →∞ p 1 m so, R = − e R as a ring direct product. Note that G (Z/p Z) = n− j=0,j:odd j n Fn/Q ∼= × pn pn µp 1 Γ/Γ for Γ = 1+ pZp as Zp× = µp 1 Γ. Thus Rn− = R0− Zp Zp[Γ/Γ ] = − − p 1× Q pn ×pn pn ⊗ pn − j=0,j:odd ejR0 Zp Zp[Γ/Γ ]; so, ejRn = Zp[Γ/Γ ]. Since Γ/Γ is a p-group, Zp[Γ/Γ ] ⊗ n ∼ is a local ring with Z [Γ/Γp ]/(γ 1) = Z as rings for the generator γ =1+ p of Γ. Q p − ∼ p To show cyclicity, we need to show that M := eiAn− is generated by one element pn over Λn := Zp[Γ/Γ ]. This is equivalent to M/mnM ∼= Fp for the maximal ideal m = (p, γ 1) Λ by Nakayama’s lemma. Let L /F be the maximal p-abelian n − ⊂ n n n extension unramified everywhere. Let Xn = GLn/Fn . We have a following field diagram F L F L n −−−→ 0 n −−−→ n x x F0 L0. −−−→ Since Fn/F0 is fully ramified at p, Fn and L0 is linearly disjoint. Thus GLn/F0 = GFn/F0 n Xn identifying G with the inertia subgroup at p of G . Therefore, G Fn/F0 Ln/F0 L0Fn/Fn ∼= GL0/F0 , and we have an exact sequence 1 G X X 1. → Ln/L0Fn → n → 0 → Since X0 is the maximal abelian quotient of GLn/F0 , GLn/L0F0 is the commutator sub- group of GLn/F0 . Since GLn/F0 = GFn/F0 n Xn and GFn/F0 is generated by γ, any j element in GLn/F0 is of the form γ x for x Xn uniquely. Then the commutator γ 1 ∈ subgroup is generated by (γ,x) = x − = (γ 1)x for x X since X is abelian. − ∈ n n In other words, GLn/F0 = (γ 1)Xn written additively as Λn-module. This implies e X /(γ 1)e X = e X and− hence e X /m e X = e X /m e X = F . Thus by i n − i n ∼ i 0 i n n i n i 0 0 i 0 ∼ p Nakayama’s lemma, eiXn is cyclic over Λn, and hence Xn is cyclic over Rn. In partic- ular, if i is even, by Kummer–Vandiver, eiX0 = 0, and hence eiXn = 0. This implies + A = 0. Therefore A− = X = R−/a for an ideal a s−. n n ∼ n n n n ⊃ n We now prove an = sn. By the lemma following this theorem Lemma 6.3, we have [R− : s−]= A− = X =[R : a ], we conclude a = s . n n | n | | n| n n n n Remark 9.4. There is another proof of this theorem via the class number formula of + Fn we describe Section 13 with more input from Kummer theory. For the proof, see [ICF, 10.3]. § ELEMENTARY IWASAWA THEORY 20 10. Iwasawa theory pn Let R = lim Rn = Λ Zp Zp[µp 1] for Λ = lim Λn = lim [Γ/Γ ]. Since Λn is a ∞ n ⊗ − n n local ring with←− maximal ideal m = (p, γ 1), Λ←− is a local←− ring with unique maximal Λn − ideal mΛ =(p, γ 1). Set X = lim Xn for Xn := GL /F = An, which is a R -module. − n n n ∼ ∞ ←− Lemma 10.1. Under the projection GF /Q GF /Q sending σ GF /Q to σ G , n0 → n ∈ n0 | Fn/Q θ−0 Q[G ] for n0 > n projects down to θ− Q[G ]. n ∈ Fn0 /Q n ∈ Fn/Q Proof. Define Hurwitz zeta function by ∞ 1 ζ(s,x)= (Re(s) > 1, 0 Lemma 10.3. We have Λ ∼= Zp[[T ]] by sending γ to t := 1+ T , where Zp[[T ]] is the one variable power series ring. ELEMENTARY IWASAWA THEORY 21 n n Proof. We have Λ = lim Z [G ] = lim Z [t]/(tp 1) as G = Γ/Γp is a cyclic n p Fn/F0 ∼ n p Fn/F0 n − group of order p generated←− by γ = σ1+←−p. Inside Z [[T ]], tpn 1 = Φ (t) for the minimal polynomial Φ (t) of pj -th roots of p − j j j unity. Since α < 1 for α := ζ n 1, any power series f(T ) Z [[T ]] converges at α ; | n|p nQ p − ∈ p n so, f(t) f(α ) gives a onto algebra homomorphism Z [[T ]] Z [µ n ] whose kernel is 7→ n p → p p generated by Φn(t). Thus (Φ (t)) m ; so, n ⊂ Λ n (tp 1) = (Φ Φ ) mn = (0). − 1 ··· n ⊂ Λ n n n \ \ \ This shows the desired result. × j For each character χ : GF∞/Q = Zp× Qp with χ µp−1 = ω− for µp 1 Zp× factoring ∼ − through G , we have χ(L ) = χ(e θ→) = L(0,χ).| Since χ(T )= χ(t)⊂ 1 = χ(γ) 1 Fn/Q j j n − − with χ(γ) 1 p < 1 as χ(γ) µp∞ = j µpj , we find χ(Lj) = Lj(χ(γ) 1) regarding | − | ∈ − k Lj as a power series Lj (T ) Zp[[T ]]. Actually we can show that Lj ((1 + p) 1) = k 1 ∈ S − (1 p − )ζ(1 k) for all integers k j +1 mod(p 1) (see [ICF, Theorem 5.11] or − − ≡ − s [LFE, 3.5 and 4.4]). The p-adic analytic function Zp s Lj((1 + p) 1) Zp is called the§ Kubota–Leopoldt§ p-adic L-function. 3 7→ − ∈ Here is a general theory of Λ-modules (see [ICF, 13.2]). If M is finitely generated tor- § sion Λ-module, then there exists finitely many non-zero elements fj Λ(j = 1, 2,...,r) and a Λ-linear map i : M r Λ/(f ) such that ker(i) < and∈ Coker(i) < → j=1 j | | ∞ | | ∞ (i.e., i is a pseudo isomorphism). Moreover the set of ideals (f ),..., (f ) is indepen- L 1 r dent of the choice of i, and the ideal char(M) := ( j fj)) is called the characteris- tic ideal of M. It is easy to see that X(j) is a torsion Λ-module of finite type as (j) (j) (j) Q (j) X1 = X /(γ 1)X Cl1− is finite. Iwasawa conjectured that char(X )=(Lj) in general for odd− j, and⊂ it was first proven by Mazur–Wiles in 1984 [MW] and there is another more elementary proof by Rubin (see [ICF, 15.7]). But the above theorem tells more that X(j) for odd j is cyclic over Λ, and Iwasawa§ conjectured also that r 1 (pseudo-cyclicity conjecture), which is not known yet. Iwasawa himself seems to have≤ had a belief not just r = 1 but the cyclicity without finite error (see [U3, C.1]). 11. An asymptotic formula of A− | n | We would like to prove the following theorem of Iwasawa: Theorem 11.1. There exist integer constants λ, µ, ν such that λn+µpn+ν A− = p | n | for all n sufficiently large. There is another theorem by Ferrero–Washington [FW] (see also [S]) which was con- jectured by Iwasawa when he proved the above theorem: Theorem 11.2. We have µ = 0. ELEMENTARY IWASAWA THEORY 22 In this section, we prove Theorem 11.1 under the Kummer–Vandiver conjecture. A n n 1 polynomial P (T ) in Zp[T ] is called distinguished if P (T ) = T + an 1T − + + a0 with p a for all i. We first quote Weierstrass’ preparation theorem in− our p-adic··· setting: | i i Theorem 11.3 (Weierstrass preparation theorem). Let f(T ) = i∞=0 aiT Zp[[T ]], i ∈ and suppose (f(T ) mod pZp[[T ]]) = i n aiT Fp[[T ]] with an = 0. Then there exists ≥ ∈ 6 a distinguished polynomial P (T ) of degree n and a unit U(T ) Z [[PT ]] such that f(T )= P ∈ p P (T )U(T ). More generally, for any non-zero f(T ) Zp[[T ]], we can find an integer µ 0 and a distinguished polynomial P (T ) and a unit∈ such that f(T )= pµP (T )U(T ). The≥ triple P (T ), µ, U(T ) is uniquely determined by f(T ). Once the first assertion is proven, P (T )= (T α) for zeros α Q with α < 0; α − ∈ p | |p so, the decomposition is unique. In other words, Spec(λ)(Q ) = α Q : α < 1 (the Q p ∼ { ∈ p | |p } open unit disk) which contains a “natural” Zp-line pZp. Before proving this theorem, we state a division algorithm for Λ: i Proposition 11.4 (Euclidean algorithm). Let f(T )= i∞=0 aiT Zp[[T ]], and suppose i ∈ (f(T ) mod pZp[[T ]]) = i n aiT Fp[[T ]] with an = 0. Suppose that the index n is the ≥ ∈ 6 P minimal with an = 0. For each g(T ) Λ, there exists a pair (q(T ),r(T )) with q(T ) Λ and r(T ) Z [T ]6 (a polynomial)P such∈ that deg(r(T )) < n and g(T )= q(T )f(T )+ r(∈T ). ∈ p n Proof. If g = 0, we have qf + r = 0. Since f has leading term anT , we find r = 0. Thus qf = 0; so, q = 0. Dividing by p and repeating this argument, we find q r 0 mod pj for all j > 0; so, q = r = 0. This shows the uniqueness of (q,r) for g =≡ 0. ≡ n 6 j Define R : Λ Λ removing first n terms and dividing by T ; so, R( j∞=0 ajT ) = j n → n ∞ a T . We put A := Id T R; so, A projects a power series to the first n-term j=n j − P up to degree n 1. Thus we have− P − (1) R(T nh(T )) = h(T ); (2) R(h) = 0 h is a polynomial of degree < n. 1 ⇔1 1 1 j j j Since R(f)− = a− (1 + Tφ(T ))− = a− ∞ ( 1) T φ(T ) Λ for φ(T ) Z [[T ]], we n n j=0 − ∈ ∈ p have R(f) Λ×. We like to∈ solve g = qf + r. This is toP solve R(g) = R(qf) by (2) above. Note that f = A(f)+ T nR(f); so, we need to solve R(g)= R(qA(f)) + R(qT nR(f)) = R(qA(f)) + qR(f) by (1) above. Write X = qR(f). Then the above equation becomes A(f) A(f) R(g)= R(X )+ X = (id+R )(X). R(f) ◦ R(f) We need to solve this equation of X. As a linear map, R A(f) has values in m as A(f) ◦ R(f) Λ is divisible by p. Thus the map id +R A(f) : X X + R(X A(f)) is invertible with ◦ R(f) 7→ R(f) inverse given by ∞ A(f) 1 A(f) j (id+R )− = ( R ) . ◦ R(f) − ◦ R(f) j=0 X ELEMENTARY IWASAWA THEORY 23 A(f) 1 1 In particular, X = (id+R )− (R(g)), and hence q = XR(f)− and r = g qf. ◦ R(f) − Once an Euclidean algorithm is known, it is a standard to have a unique factorization theorem from the time of Euclid: Corollary 11.5. The ring Λ is a unique factorization domain. µ Proof of Weierstrass theorem: Dividing f(T ) by p for µ = minj v(aj) for the p-adic valuation v, we may assume that an = 0 for n as above. Therefore we only prove the first part. Apply Euclidean algorithm6 above to g = T n, we get T n = qf + r with a n polynomial r of degree < n. Writing q(T )= j∞=0 cnT and comparing the coefficient n 1 n 1 of T , we get 1 c a mod m ; so, c Z ×; so, f = q (T r). Thus U(T ) := q 0 n Λ 0 p P − − and P (T ) := T n≡ r satisfies the property∈ of the preparation theorem.− − Lemma 11.6. If E =Λ/pµΛ, then E/(γpn 1)E = pµpn . | − | µ µ µ pn µ Proof. Since E = Λ/p Λ=(Z/p Z)[[T ]], from (Z/p Z)[[T ]]/(γ 1)(Z/p Z)[[T ]] ∼= (Z/pµZ)[[T ]][Γ/Γpn], we conclude the desired assertion. − Lemma 11.7. If E =Λ/g(T )Λ for a distinguished polynomial of degree λ with g(ζ 1) = − 6 0 for all ζ µp∞ , then there exists an integer n0 > 0 and a constaint ν Z such that E/(γpn 1)∈E = pλn+ν for all n > n . ∈ | − | 0 λ Proof. A monic polynomial f(T ) Zp[T ] is distinguished if and only if f(T ) T mod p; so, a product and a factor of distinguished∈ polynomials are distinguished. ≡ n 0 0 γp 1 pn pn jpn λ We put Nn,n0 := 0− = − γ for n > n0. Writing g(T )= T pQ(T ) with γpn 1 j=0 − − λ k Q(T ) Zp[T ], we have T PpQ(T ) mod g; so, T pQk(T ) mod g for all k λ with ∈ ≡ n ≡ ≥ some polynomial Qk(T ) Zp[T ]. Therefore if p > λ, n ∈ n n γp = (1+ T )p =1+ pR(T )+ T p 1+ pS (T ) mod g(T ) ≡ n for R(T ),S (T ) Z [T ]. Thus n ∈ p pn+1 p 2 (1 + T ) (1 + pS (T )) 1+ p S0 (T ) mod g(T ). ≡ n ≡ n Thus we find pn+2 pn+1 p pn+1 pn+1 (p 1)pn+1 γ 1=(γ ) 1=(γ 1)(1 + γ + + γ − ) − − − ··· p n+1 n+1 (1+ +1+p2P (T ))(γp 1) mod g(T )= p(1 + pP (T ))(γp 1), ≡ ··· − − 1 j where P (T ) zZp[T}|]. Since{ (1+pP (T ))− = j∞=0( pP (T )) Λ, 1+pP (T ) is a unit in ∈ n+2 − ∈ γp 1 Λ. Therefore multiplication by N = n+1 − is equal to multiplication by p on E n+2,n+1 γPp 1 − λ n0 as long as E is Zp-free of rank λ; so, we find E/pE = p . Thus if p > λ and n n0, we get the desired formula. | | ≥ (j) Since X = Λ/(Lj ) for j odd with j > 1, by the above two lemmas, we get Theo- µj rem 11.1 for λ = 1 Remark 11.8. Actually for any Zp-extension K /K = n Kn/K (i.e., GK∞/K = Zp and n ∞ ∼ GK∞/Kn = p Zp) for a number field K, writing the p-primary part of the class number ∼ n p Zp en nS of Kn = K as p , it is known that en = ln + mp + c for constants l,m,c if n is ∞ sufficiently large (see [ICF, Theorem 13.3]). If is a smooth projective curve over the finite field F , we can think of the extension X p Fp( )/Fp( ) of the function fields, where Fp is a fixed algebraic closure of Fp. Then X X Z Gal(F ( )/F ( )) = Z = Frobb. Put X = Gal(L/F ( )) for the maximal abelian p X p X ∼ p p X extension unramified everywhere L/F ( ). Then Gal(F ( )/F ( )) acts on X by con- p X p X p X jugation. Let /Fp be theb Jacobian variety of (i.e., degree 0 divisors modulo principal divisors). ThenJ X = T for the Tate moduleX T = lim [ln] for primes l. In l lJ lJ n J particular, T Z2g for the genus g of if l = p. If l ←−= p, then T Z r for l ∼= l Q p ∼= p 0 r g calledJ the p-rank of . The FrobeniusX 6 has reciprocal characteristicJ polyno- ≤ ≤ X mial det(12g Frobp Tl x) = φ(x) Z[x] independent of l = p. Then (the main part − | J ∈ s 6 of) the zeta function of is given by L(s, ) = φ(p− ). For an eigenvalue α of Frobp X X s on Tl , Weil proved that α = √p. Thus by definition, L(s, ) = 0 p− = α for an J | | 1 X ⇔ eigenvalue α. This implies Re(s)= 2 (Riemann hypothesis for ). In Iwasawa’s case, for the maximal p-abelian extension L/F Xunramified everywhere, Z∞p X := Gal(L/F ) is a module over Γ := Gal(F /F0) = γ . By µ = 0, X is Zp- free (under the∞ Kummer–Vandiver conjecture). We∞ have an isomorphism of Λ-modules X = 0 12. Cyclotomic units + We now prepare some facts for determining Cln as an index of the cyclotomic units + | | in the entire units in On . This is a base of the proof of the cyclicity theorem in [ICF, r s 10.3] a bit different from our proof in Section 9. For a number field, if F Q R ∼= R C , § (pn⊗+1 pn)/2 × we know rankZ O× = dimQ O× Z Q = r + s 1. Since Fn Q R = C − , we find n+1 n ⊗ − ⊗ rank On× =(p p )/2 1 for the integer ring On of Fn. Let Vn be the multiplicative − − a n+1 group generated by µ n+1 1 ζ n+1 1 cyclotomic unit. Lemma 12.1. (1) C+ := C F + is generated by 1 and the units n n ∩ n − a 1 ζ n+1 (1 a)/2 p n+1 − − ξa := ζpn+1 with 1 Proof. Let m = n + 1, and write ζ := ζpm . Note that a a/2 a/2 m (1 a)/2 1 ζ ζ− ζ sin(πa/p ) ξ = ζ − − = − = . a 1 ζ ζ 1/2 ζ1/2 ± sin(π/pm) − − − 1/2 + σa a Since ζ = ζpm as p is odd, ξa Fm . Note that (1 ζ) = p = p = (1 ζ ) for − ∈ − − + the unique prime ideal p of Z[µpm ] above p, we find (ξa)= Om, and hence ξa (Om)×. Thus the assertion (2) implies (1). ∈ pm−k We now prove (2). Note that ζ generates µ k ; so, we have for 0 k Theorem 13.1 (R. Dedekind). We have r s 2 (2π) ClF RF Ress=1ζF (s)= lim (s 1)ζF (s)= | | , s +1 − w D → | F | where w is the number of roots of unity in F and DF is thep discriminant of F . Theorem 13.2 (Dirichlet/Hecke). The L-functions L(s,χ) and ζF (s) can be continued analytically to C 1 and satisfies −{ } 1 Λ (s)ζ (s)=Λ (1 s)ζ (s) and Λ (s)L(s,χ)= ε Λ (1 s)L(1 s,χ− ), F F F − F χ χ χ − − s s r s s [F :Q]/2 G(χ) where ΛF (s)= A Γ( ) Γ(s) with A = 2− π− DF , εχ = − for the conduc- 2 √χ( 1)f | | − s/2 s+δ 1 χ( 1) − − tor f of χ and Λχ(s)=(f/π) Γ( 2 ) with δ = p2 . If χ = 1, L(s,χ) is holomorphic everywhere on C. 6 Since Ress=1ζ(s) = 1, we get Corollary 13.3. We have ζ (s) 2r(2π)s Cl R lim F = | F | F . s +1 ζ(s) w D → | F | Lemma 13.4. Let η1,...,ηR be a basis of a subgroupp E of O× modulo torsion. Then we have { } RF =[O×/torsion : E]. RF (η1,...,ηR) This is because RF (η1,...,ηR) is the volume of the lattice spanned by Log(η) = σj di (log η )i in the subspace x F TrF/Q(x) = 0 Q R of the real vector space F Q R. See [ICF,| | Lemma 4.15] for more{ ∈ details.| }⊗ ⊗ + Suppose now that F = Fn with m = n +1 for n 0. We now quote from [ICF, Theorem 4.9]: ≥ m Theorem 13.5 (Dirichlet–Kummer). For a primitive character χ : (Z/p Z)× Q× with χ( 1) = 1, we have → − pm τ(χ) a L(1,χ)= χ(a) log 1 ζ m . − pm | − p | a=1 X m The Galois group G = (Z/p Z)×/ 1 acts on F , and hence F = Q Ker(Tr ) F/Q ∼ {± } ⊕ F/Q as a Galois module. Let ρ be the representation of GF/Q on Ker(TrF/Q). By a normal m basis theorem of Galois theory, we have F ∼= Q[GF/Q] ∼= Q[(Z/p Z)×] as Galois modules; m so, F Q = χ, where χ runs over all characters of (Z/p Z)× with χ( 1) = 1. ⊗Q ∼ χ − Thus shows ρ = χ=1 χ. Then we have ∼L 6 Lemma 13.6. We have ζ + (s)= ζ(s)L(s, ρ)= ζ(s) L(s,χ), where χ runs L Fn χ=1,χ( 1)=1 m 6 − over all characters of (Z/p Z)×/ 1 and L(s,χ) is the Dirichlet L-function of a char- m {± } Q acter χ :(Z/p Z)×/ 1 Q×. {± }→ ELEMENTARY IWASAWA THEORY 27 g Proof. Let 0 + Corollary 13.7. For F = Fn , we have 2[F :Q] Cl R L(1,χ)= | F | F , χ=1 w DF Y6 | | n+1 where χ runs over all non-trivial characters of (Zp/p Z)× with χ( 1) = 1. In addition, we have − (13.1) ε = 1 and ( G(χ)) = D . χ − | F | χ=1 χ=1 Y6 Y6 p This follows from Lemma 13.6 and the functional equation of Theorem 13.2 via the s s+1 1 s formula Γ( 2 )Γ( 2 ) = 2 − √πΓ(s). Exercise 13.8. For a general field F not necessarily abelian over Q, we write the Galois representation on V := Ker(Tr ) F as ρ and define F/Q ⊂ s 1 L(s, ρ)= det(1 ρ I (Frob )l− )− (Artin L-finction of ρ). − |V l l Yl Prove that ζF (s)= ζ(s)L(s, ρ). Lemma 13.9. Let G be a finite abelian group, and let f : G C be a function. Then 1 → (1) det(f(τσ− ))σ,τ G = det(f(στ))σ,τ G = × χ(σ)f(σ); ∈ ∈ χ Hom(G,C ) σ G 1 ∈ ∈ (2) det(f(τσ− ) f(τ))σ,τ=1 = det(f(στ) f(τ))σ,τ=1 = χ=1 σ G χ(σ)f(σ). − 6 −Q 6 P 6 ∈ A proof of this lemma will be given after proving the followinQ g theorem:P + + + + Theorem 13.10. m Let F = Fn . Then we have ClFn = [(On )× : Cn ], and ξa 1 a + Note that ξa = [F : Q] 1 for F = Fn . Thus we can think of the regulator |{ }| − 1/2 σ σ RC := RF ( ξa ). Write ζ = ζpm (m = n+1) and l(σ) = log (ζ− (1 ζ)) = log 1 ζ for σ G ={ G} . We have for G = G | − | | − | ∈ F/Q F/Q τ RC = det(log ξa )a,τ G 1 ± | | ∈ −{ } = det(l(στ) l(τ))σ,τ=1 ± − 6 Lemma 13.9 (2) = χ(σ)l(σ) ± χ=1 σ G Y6 X∈ = χ(a) log (1 ζ)σa ± | − | χ=1 1 a 1 Thus the matrix expression of T with respect to the basis β is given by (f(τσ− ))σ,τ G. ∈ Since χ χ Hom(G,C×) also form an T -eigen basis of V with eigenvalue σ χ(σ)f(σ). This shows{ (1).} ∈ P To show (2), let V0 = h V s h(σ) = 0 which is stable under the action of 1 { ∈ | } G. Set ψσ = φσ G which is in V0, and β0 := ψσ σ=1 forms a basis of V0. Since − | | P { } 6 ψ1 + σ=1 ψσ = 0, we have ψ1 = τ=1 ψτ . Since 6 − 6 1 ψ (σx) = 1 (1/ G ) τx = τ x = σ− τ φ −1 (x), P τ − | | P⇔ ⇔ ⇔ σ τ we have 1 1 Tψ (x)= f(τσ− )φ (x)= (f(τσ− ) f(τ))φ (x). τ σ − σ σ G σ=1 ∈ 6 X 1 X Then T V has the matrix expression (f(τσ− ) f(τ))σ,τ=1. This shows (2). | 0 − 6 References Books [BNT] A. Weil, Basic Number Theory, Springer, New York, 1974. [CFT] J. 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