ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2
MATTHIAS FLACH
Let l be prime number, m0 an integer prime to l and n × Λ = lim Z [Gal(Q(ζ n )/Q)] =∼ lim Z [(Z/m l Z) ] ←− l m0l ←− l 0 n n the cyclotomic Iwasawa algebra of ”tame level m0”. Using ´etalecohomology one can define a certain perfect complex of Λ-modules ∆∞ (see section 1.2 below) and a ∞ certain basis L of the invertible Q(Λ)-module DetQ(Λ)(∆ ⊗Λ Q(Λ)) where Q(Λ) is the total ring of fractions of Λ. This basis L is obtained by l-adically interpolating the leading Taylor coefficients of the Dirichlet L-functions L(χ, s) at s = 0 where χ ∞ runs through characters of conductor dividing m0l . The main conjecture referred to in the title of this paper, Theorem 1.2 below, is the statement that L is in fact ∞ a Λ-basis of DetΛ ∆ . This main conjecture was essentially proven for l 6= 2 by Burns and Greither [7] building on the theorem of Mazur and Wiles [15] (see also Rubin [18]) proving the traditional ”Iwasawa main conjecture”. The extra refinement in the theorem of Burns and Greither vis-a-vis the theorem of Mazur and Wiles lies in the fact that Λ need not be a regular ring. Indeed if l is odd then Λ is regular if and only if l - φ(m0) where φ is the Euler φ-function. In this article we give a proof of Theorem 1.2 for l = 2. This was claimed as Theorem 5.2 in the survey paper [9] but the proof given there, arguing separately for each height one prime q of Λ, is incomplete at primes q which contain l = 2. The argument given in [9][p.95] is an attempt to use only knowledge of the cohomology ∞ as well as perfectness of the complex ∆q but it turns out that this information is insufficient. Here we shall use the techniques of the paper [6], such as the Coleman homomorphism and Leopoldt’s result on the Galois structure of cyclotomic integer ∞ rings, to construct the complex ∆q more explicitly and thereby verify Theorem 1.2. What is peculiar to the situation l = 2 is not only that Λ is never regular (due to the presence of the complex conjugation) but also that the l-adic L-function L is quite differently defined for even and odd characters. For even characters one interpolates first derivatives of Dirichlet L-functions via cyclotomic units, for odd characters one interpolates the values of these functions via Stickelberger elements. A proof of Theorem 1.2 therefore in some sense involves a ”mod 2 congruence” between Stickelberger elements and cyclotomic units. Given the explicit nature of both objects it is perhaps not surprising that this congruence turns out to be an elementary statement which is arrived at, however, only after some rather arduous computations. The statement is the following. Let M ≡ 1 mod 4 be an integer,
2000 Mathematics Subject Classification. Primary: 11G40, Secondary: 11R23, 11R33, 11G18. The author was supported by grant DMS-0401403 from the National Science Foundation. 1 2 MATTHIAS FLACH
0 < x < 1 a real number and u = exp(2πix). Then the sign of the real number 1−uM bMxc (1−u)M is (−1) . We believe that Theorem 1.2 (for any prime l) is the most precise statement one can make about the relation between leading Taylor coefficients of Dirichlet L-functions at s = 0 and ´etalecohomology. It is a special case of very general conjectures on motivic L-values put forward by Kato [14] and Kato and Fukaya [10]. Theorem 1.2 for l = 2 not only confirms Kato’s point of view of L-values and l-adic L-functions as bases of determinant line bundles quite beautifully but also has other number theoretic consequences which were already noted in [9][5.1] and [6][Cor. 1.2-1.4]. In particular it completes the proof of the equivariant Tamagawa number conjecture for the motive h0(Spec(L))(j) and the order Z[Gal(L/Q)] where j is any integer and L/Q any abelian extension. This includes the 2-primary part of the original (non-equivariant) Tamagawa number conjecture of Bloch and Kato [1] for the Riemann Zeta function. A consequence of the equivariant Tamagawa number conjecture is the validity of all three Chinburg conjectures asserting the vanishing of certain invariants Ω(L/K, i) = 0, i = 1, 2, 3 if L/Q is abelian (see [6][Cor. 1.4]). For other consequences concerning Fitting ideals we refer to Greither’s paper [12] and for the relevance to conjectures of Tate, Stark, Gross, Rubin, Popescu et al to Burns’ paper [3][Thm A]. After recalling material from [9] and [6] in section 1 we prove in section 2 a statement which might be regarded as a functional equation of L and which is a refinement of a result going back to Iwasawa. We finally give the proof of Theorem 1.2 in section 3 and collect some technical computations related to the Shapiro Lemma in an Appendix.
1. Notation and Preliminaries 1.1. Cyclotomic fields. We follow the notation of [9] which we now recall. For any integer m we set 2πi/m ζm :=e ,Lm := Q(ζm)
σm :=the inclusion Lm → C ∼ × Gm := Gal(Lm/Q) = (Z/mZ) . × For any Dirichlet character η : Gm → C we denote by X 1 −1 eη = eη,m = η(a) τa,m |Gm| a∈(Z/mZ)× a the corresponding idempotent where τa,m ∈ Gm is defined by τa,m(ζm) = ζm.
Lemma 1.1. For any Q-rational (resp. Ql-rational) character χ of Gm of conduc- a tor fχ | m, any d | m and any primitive d-th root of unity ζd we have 0 if fχ - d (1) e ζa = φ(f ) d d χ d χ −1 µ( )χ ( )χ(a)eχζfχ if fχ | d φ(d) fχ fχ in Lm (resp. Lm ⊗Q Ql). Here φ(m) is Euler’s φ-function, µ(m) is the M¨obius function, χ(a) = 0 if (a, fχ) > 1 and we view a Q-rational character χ as the tautological homomorphism × × × Gm → Q[Gm] → (eχQ[Gm]) = Q(χ) ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 3 P where eχ = η7→χ eη (and similarly for Ql-rational characters). Proof. This is Lemma 6.2 of [6]. ¤
For a prime number l and integer m0 prime to l we define ( l l 6= 2 ` = 4 l = 2
G ∞ := lim G n m0l ←− m0l n
∞ γ :=1 + `m0 ∈ Gm0l
Λ := lim Z [G n ] =∼ Z [G ][[γ − 1]]. ←− l m0l l m0` n The Iwasawa algebra Λ is a finite product of complete local 2-dimensional Cohen- Macaulay (even complete intersection) rings. However, it is regular if and only if l - #G`m0 . We denote by Q(Λ) the total ring of fractions of Λ.
1.2. Global Iwasawa Theory. Borrowing notation from [6] (and replacing p by l and setting r = 1) we define a free, rank one Λ-module with a continuous Λ-linear action of Gal(Q¯ /Q)
∞ 0 T = lim H (Spec(L n ⊗ Q¯ ), Z (1)) l ←− m0l Q l n and a perfect complex of Λ-modules
∞ 1 ∞,∗ ∗ ∆ = RΓc(Z[ ],Tl (1)) [−3] m0l where for any projective Λ-module P (resp. perfect complex of Λ-modules C) ∗ ∗ we set P = HomΛ(P, Λ) (resp. C = R HomΛ(C, Λ)). The compact support ´etalecohomology is defined as in [5][p. 522]. We recall the computation of the cohomology of ∆∞ from [9]. We have Hi(∆∞) = 0 for i 6= 1, 2, a canonical isomorphism 1 ∞ ∼ ∞ 1 × H (∆ ) = U := lim OL n [ ] ⊗Z Zl {v|m0l} ←− m0l n m0l and a short exact sequence
0 → P ∞ → H2(∆∞) → X∞ → 0 {v|m0l} {v|m0l∞} where
∞ 1 P := lim Pic(OL n [ ]) ⊗Z Zl {v|m0l} ←− m0l n m0l ∞ X := lim X{v|m l∞}(Lm ln ) ⊗Z Zl. {v|m0l∞} ←− 0 0 n For any number field L and set of places S of L we set here M X YS(L) := Z,XS(L) := {(av) ∈ YS(L)| av = 0} v∈S v 4 MATTHIAS FLACH and all limits are taken with respect to Norm maps (on YS this is the map sending a place to its restriction). For d | m0 put ∞ (2) η :=(1 − ζ n ) ∈ U d `dl n≥0 {v|m0l} ∞ σ :=(σ n ) ∈ Y `m0l n≥0 {v|m0l∞}
1 −1 θd :=(g`dln )n≥0 ∈ · (γ − χcyclo(γ)) Λ [Lm0 : Ld] where µ ¶ X a 1 (3) g = − − τ −1 ∈ Q[G ] k k 2 a,k k 0 a with τa,k ∈ Gk defined by τa,k(ζk) = ζk . Here we also view τa,k as an element of 0 Q[Gk0 ] for k | k (which allows us to view θd as an element of the fraction field of −1 P Λ for d | m0) by τa,k 7→ [Gk0 : Gk] a0≡a mod k τa0,k0 . The relationship between θd and l-adic L-functions in the usual normalization is given by the interpolation formula j −1 −j −1 −1 1−j (4) χχcyclo(θd) = (1 − χ (l)l )L(χ , j) =: Ll(χ ω , j) n for all Dirichlet characters χ of conductor dl and j ≤ 0 (here ω and χcyclo denote × ∞ the Teichmueller and cyclotomic character Gm0l → Zl , respectively). We fix an embedding Q¯ l → C and identify Gˆk with the set of Q¯ l-valued charac- ters. The total ring of fractions Y (5) Q(Λ) =∼ Q(ψ) Q ψ∈Gˆ l `m0 of Λ is a product of fields indexed by the Ql-rational characters ψ of G`m0 . As in [9] one easily computes ( ∞ ∞ 1 ψ even dimQ(ψ)(U ⊗Λ Q(ψ)) = dimQ(ψ)(Y ⊗Λ Q(ψ)) = {v|m0l} {v|m0l∞} 0 ψ odd. Note that the inclusion X∞ ⊆ Y ∞ becomes an isomorphism after {v|m0l∞} {v|m0l∞} −1 tensoring with Q(ψ) and that eψ(ηm0 ⊗ σ) is a Q(ψ)-basis of Det−1 (U ∞ ⊗ Q(ψ)) ⊗ Det (X∞ ⊗ Q(ψ)) Q(ψ) {v|m0l} Λ Q(ψ) {v|m0l∞} Λ ∼ ∞ =DetQ(ψ) (∆ ⊗Λ Q(ψ)) ∞ for even ψ. For odd ψ the complex ∆ ⊗Λ Q(ψ) is acyclic and we can view eψθm0 ∈ Q(ψ) as an element of ∞ ∼ DetQ(ψ) (∆ ⊗Λ Q(ψ)) = Q(ψ). −1 Note also that eψθm0 = 0 (resp. eψ(ηm0 ⊗ σ) = 0) if ψ is even (resp. odd). Hence we obtain a Q(Λ)-basis −1 −1 ∞ L := θm0 + 2 · ηm0 ⊗ σ ∈ DetQ(Λ) (∆ ⊗Λ Q(Λ)) ∞ of the invertible Q(Λ)-module DetQ(Λ) (∆ ⊗Λ Q(Λ)). ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 5 Theorem 1.2. There is an equality of invertible Λ-submodules ∞ Λ ·L = DetΛ∆ ∞ of DetQ(Λ) (∆ ⊗Λ Q(Λ)). For odd primes l this theorem is essentially due to Burns and Greither [7] and a proof of this precise statement with this precise notation was given in [9][Sec. 5.1]. For l = 2 the theorem is new and its proof will occupy section 3 of the present paper. 1.3. Local Iwasawa Theory. Here we recall results from [6]. The cohomology of ∞ RΓ(Ql,Tl ) is naturally isomorphic to × n (lim Q(ζ n ) /l ) ⊗ Z i = 1 ←−n m0l l Zl l i ∞ ∼ Q (6) H (Ql,Tl ) = v|l Zl i = 2 0 otherwise n n where the limit is taken with respect to the norm maps (and Q(ζm0l )l = Q(ζm0l )⊗Q Ql denotes a finite product of local fields). The valuation map induces a natural short exact sequence Y ∞ × n × n val n (7) 0 → Uloc := lim O n /l → lim Q(ζm0l )l /l −−→ Zl → 0 ←− Q(ζm0l )l ←− n n v|l and in addition Coleman has constructed an exact sequence [17, Prop. 4.1.3] Y Y ∞ Θ (8) 0 → Zl(1) → Uloc −→ R → Zl(1) → 0 v|l v|l where X R := {f ∈ Z[ζm0 ]l[[X]] | ψ(f) := f(ζ(1 + X) − 1) = 0} ζl=1 and Z[ζm0 ]l denotes the finite ´etale Zl-algebra Z[ζm0 ] ⊗Z Zl. Moreover the map Θ is given by µ ¶ φ (9) Θ(u) = 1 − log(f ) l u where fu is the (unique) Coleman power series of the norm compatible system of Fr l units u with respect to (ζln )n≥1 and one has φ(f)(X) = f l ((1 + X) − 1). The ∞ Zl-module R carries a natural continuous Gm0l -action [17, 1.1.4], and with respect to this action all maps in (6), (7) and (8) are Λ-equivariant. Lemma 1.3. The Λ-module R is free of rank one with basis X ∞ βm0 := ξm0 (1 + X); ξm0 := ζd m1|d|m0 Q where m = p. 1 p|m0 Proof. This is Lemma 5.1 of [6]. ¤ ∞ Proposition 1.4. Viewing βm0 as a Q(Λ)-basis of ∼ ∞ ∼ 1 ∞ R ⊗Λ Q(Λ) = Uloc ⊗Λ Q(Λ) = H (Ql,Tl ) ⊗Λ Q(Λ) ∼ −1 ∞ = (DetΛ RΓ(Ql,Tl )) ⊗Λ Q(Λ), 6 MATTHIAS FLACH one has ∞ −1 ∞ −1 ∞ Λ · βm0 = DetΛ RΓ(Ql,Tl ) ⊂ (DetΛ RΓ(Ql,Tl )) ⊗Λ Q(Λ). Proof. This is Proposition 5.2. of [6]. ¤ The following proposition is an application of the reciprocity law of Coleman [8][Thm. 1] and is not contained in [6]. Our proof follows very closely Perrin- Riou’s proof of her reciprocity law [17][Conj. 3.6.4] in the special case of an l-adic representation V = V0(1), V0 unramified, in [17][Thm. 4.3.2]. Since the emphasis in [17] is on a general crystalline representation and the prime l = 2 is systematically excluded we prefer to give the details of this argument again. The cup-product pairing ∞ ∞,∗ RΓ(Ql,Tl ) × RΓ(Ql,Tl (1)) → RΓ(Ql, Λ(1)) → Λ[−2] induces a pairing ∞ L ∞,∗ RΓ(Ql,Tl ) ⊗Λ RΓ(Ql,Tl (1)) → Λ[−2] and a pairing on determinants −1 ∞ −1 ∞,∗ (10) DetΛ RΓ(Ql,Tl ) ⊗Λ DetΛ RΓ(Ql,Tl (1)) → Λ which is perfect by local Tate duality, i.e. the arrow is an isomorphism. Denote by # × ∞ κ : Gm0l → Λ the character # −1 (11) κ : g 7→ χcyclo(g)g as well as the induced ring automorphism of Λ. Note that there is a natural iso- morphism of Gal(Q¯ /Q)-modules ∞,∗ ∼ ∞ Tl (1) = Tl ⊗ Λ Λ,κ# which induces an isomorphism of perfect complexes of Λ-modules ∞,∗ ∼ ∞ RΓ(Ql,Tl (1)) = RΓ(Ql,Tl ) ⊗ Λ. Λ,κ# Hence we may regard the pairing (10) as a perfect κ#-sesquilinear pairing < −, − > −1 ∞ ∞ on DetΛ RΓ(Ql,Tl ). Recall that βm0 is a basis of this last module by Proposition 1.4. Let ∞ ∞ × (12) % :=< βm0 , βm0 >∈ Λ ∞ be the discriminant of this pairing, let c ∈ Gm0l be the complex conjugation and ∞ cl the projection of c into the direct factor G` of Gm0l . tor Proposition 1.5. The element % lies in Z [G ∞ ] = Z [G ] and we have % = l m0l l m0` P −1 %0cl where %0 = %0(g)g ∈ Zl[Gm0 ] is given by g∈Gm0 µ ¶ X φ(m ) d (13) % (g) = 0 µ , 0 φ(d/(d, g + 1)) (d, g + 1) m1|d|((g+1)m1,m0) viewing g ∈ Gm0 as an integer prime to m0. ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 7 Proof. Cup product also induces the κ#-sesquilinear pairing in the top row of the following diagram 1 ∞ 1 ∞ <−,−> H (Ql,T ) × H (Ql,T ) −−−−−→ Λ x l x l k U ∞ × U ∞ −−−−→ Λ loc loc Θy Θy k <−,−> R × R −−−−−→ Λ and Coleman’s reciprocity law will allow us to give an explicit pairing in the bottom row that makes the diagram commutative. Note that 1 ∞ 1 H (Q ,T ) =∼ lim H (Q(ζ n ) , µ n ) l l ←− m0l l l n 1 ∞ and for u, v ∈ H (Ql,Tl ) we have X g −1 n n < u, v >≡ < un, vn >n g in Z/l Z[Gm0l ] n g∈Gm0l where the Hilbert symbol 1 1 n n n n n < −, − >n: H (Q(ζm0l )l, µl ) × H (Q(ζm0l )l, µl ) → Z/l Z Since X X F (ζ − 1)D(log fv)(ζ − 1) = F (ζ − 1)D(Θ(v))(ζ − 1) ζ∈µln ζ∈µln by the argument at the bottom of page 146 of [17] we conclude that the pairing < −, − > on R given by X g −1 n n < F, G >≡ < F , G >n g in Z/l Z[Gm0l ] n g∈Gm0l where X < F, G > ≡ Tr l−n F (ζ − 1)D(G)(ζ − 1) mod ln n Q(ζm0 )l/Ql ζ∈µln n makes the above diagram commutative. Note here that the action of g ∈ Gm0l ln−1 2n−2 ∞ on F ∈ R is independent of the lift of g to Gm0l modulo γ − 1 (γ − 1 if 8 MATTHIAS FLACH ∞ l = 2). Taking F = G = βm0 = ξm0 (1 + T ) we have D(G) = G and X < β∞,g, β∞ > ≡ Tr l−n ξg ζgξ ζ mod ln m0 m0 n Q(ζm0 )l/Ql m0 m0 ζ∈µln ( 1+g n TrQ(ζ ) /Q (ξ ) if g ≡ −1 mod l ≡ m0 l l m0 0 otherwise. Now by Lemma 1.1 Tr (ξ1+g) = φ(m )e ξ1+g Q(ζm0 )l/Ql m0 0 1 m0 X 0 X φ(m ) (14) = φ(m ) e ζD/d f+gD/df = 0 µ(D/f) · 1 0 1 D/f φ(D/f) d,d0 d,d0 0 0 where d, d run through all multiples of m1 dividing m0, D = l.c.m.(d, d ) and 0 f = g.c.d.(D/d + gD/d, D). Here we view g as an integer prime to m0. Now if d 6= d0 then there exists a prime p dividing D/d0 or D/d but not both since D is the least common multiple. Hence p - f. Also, p2 | D since p | d and p | d0 in any case. Hence p2 | D/f and µ(D/f) = 0. So the sum (14) reduces to the diagonal X X 1+g φ(m0) φ(m0) TrQ(ζ ) /Q (ξ ) = µ(d/f) = µ(d/f) m0 l l m0 φ(d/f) φ(d/f) m1|d|m0 m1|d|((g+1)m1,m0) which in turn reduces to a sum over d so that d/f = d/(d, g + 1) is squarefree. This happens if and only if d | (g + 1)m1. In summary then X ∞ ∞ ∞,g ∞ −1 % =< βm0 , βm0 >= < βm0 , βm0 >n g = %0cl g∈G n m0l n where %0 was defined in (13). ¤ 2. The functional equation In this section we prove a result, Proposition 2.2 below, which is a key ingredient in the proof of Theorem 1.2 for l = 2 but which is valid for all primes l and which is a more precise form of a theorem going back to Iwasawa (”the l-adic L-function is the characteristic ideal of local units modulo cyclotomic units”). Proposition 2.2 can also be viewed as an incarnation of the functional equation of the l-adic L-function L as we shall explain at the end of this section. tor For any subgroup H ⊆ G ⊂ G ∞ we set m0 m0l X 1 X e = e = g, H χ |H| χ(H)=1 g∈H where the sum is over all Ql-rational characters that vanish on H. For any prime p 6= l we denote by tor × ∞ Ip ⊆ Gm0 ⊂ Gm0l ⊆ Λ the inertia subgroup at p. For x ∈ Λ we set 1 e (x) = 1 + (x − 1)e ∈ Λ[ ] p Ip l ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 9 ∞ so that x 7→ ep(x) is a multiplicative map. We denote by Frp ∈ Gm0l (resp. ˜ ∞ Frp ∈ Gm0l ) the choice of a Frobenius element with trivial component in Ip (resp. ∞ Gm0 ). Note here that Ip and Gm0 are canonically direct factors of Gm0l by the Chinese remainder theorem. × Lemma 2.1. Fix a prime p that is not equal toPl. If u : Z → Λ is any function such that p − 1 divides u(0) − u(1) in Λ, then χ u(ordp(fχ))eχ,m0 is a unit of Λ where the sum is over all Ql-rational characters of Gm0 . Proof. This is Lemma 4.5 of [6] with Zl[G] replaced by Λ. The proof of this Lemma in loc. cit. transfers verbatim (with the roles of l and p interchanged). ¤ × Proposition 2.2. Using Lemma 2.1 define vp ∈ Λ via the function max(x,1) x 7→ (p/ Fr˜ p) . With the notation of (2), (9), (11) and Lemma 1.3 one has à ! Y 1 ep(1 − Frp) # ∞ Θ(ηm ) = − vp κ (θm ) · β 0 Frp 0 m0 m0 ep(1 − ) p|m0 p if m0 > 1 and # ∞ Θ((γ − 1)ηm0 ) = −(γ − 1)κ (θm0 ) · βm0 if m0 = 1. Remarks. a) If Λ is regular, equivalently if l - φ(`m0), this identity, up to an unspecified unit in Λ, is essentially due to Gillard [11][Theorem 1] and goes back to Iwasawa [13], see also [19][Theorem 13.56], for m0 = 1 and l 6= 2. However, even if Λ is regular we need an exact formula in our later applications. b) Following the point of view that L is an l-adic L-function with Euler factors at primes dividing m0 removed, one might be tempted to regard the factor Y ep(1 − Frp) Frp ep(1 − ) p|m0 p as the quotient of the missing Euler factors of ηm0 and θm0 . This is however not −1 quite correct since the Euler factor of η is not 1 − Frp but 1 − Frp , i.e. the ∞ 0 −1 0 ord (m ) projection of η into U ⊗ Λ is (1 − Fr )η 0 where m = m p p 0 . m0 {v|m0l} Λ p m0 0 0 One cannot absorb this difference into the unit vp by introducing the factor − Frp = −1 × (1 − Frp)/(1 − Frp ) since ep(− Frp) ∈/ Λ . Nonetheless, the formula in Proposition 2.2 is correct. One can verify that it is compatible with projection from level m0 0 to level m0, and it is also compatible with Lemma 2.5 below. Proof. Let −1 m0 f(X) := 1 − ζm0 (1 + X) be the Coleman power series for ηm0 and define g(X) ∈ Z[ζm0 ]l[[X]] by −1 ∞ (15) g(X) := Θ(ηm0 ) = (1 − l φ) log(f(X)) = λ · βm0 where this last equality defines λ ∈ Λ via Lemma 1.3. By a standard result for the Amice transform [17][Lemme 1.1.6 (ii)] one has for k ≥ 1 k k (16) χcyclo(λ) · ξm0 = (D g)(0) ∈ Z[ζm0 ]l 10 MATTHIAS FLACH d where D is the differential operator D = (1 + X) dX . Since Dφ = lφD one has Dkg(X) = Dk(1 − l−1φ) log(f(X)) = (1 − φlk−1)Dk log(f(X)) f 0(X) = (1 − φlk−1)Dk−1(1 + X) f(X) −1 −1 m0 k−1 k−1 −ζm0 m0 (1 + X) = (1 − φl )D −1 m0 1 − ζm0 (1 + X) µ ¶k−1 d ζ m−1ez/m0 = (1 − φlk−1) m0 0 dz ζ ez/m0 − 1 Ãm0 ! µ ¶k−1 m d 1 X0 eaz/m0 = (1 − φlk−1) ζa m0 z dz m0 e − 1 a=1 à ! m0 µ ¶k−1 ∞ a n−1 1 X d 1 X Bn( ) z = (1 − φlk−1) ζa + m0 m m0 dz z n (n − 1)! 0 a=1 n=1 z d where X = e − 1, D = dz and Bn(t) ∈ Q[t] is the Bernoulli polynomial. Taking X = 0 or equivalently z = 0 we find m0 a X Bk( ) k k−1 1 a m0 (17) (D g)(0) = (1 − Frl l ) ζ . m m0 k 0 a=1 Let χ be a Ql-rational character of Gm0 and eχ = eχ,m0 the corresponding idem- potent. Then Lemma 1.1 together with Lemma 2.3 below implies m0 a X Bk( ) a m0 eχ ζ m0 k a=1 X Xd a0 a0 Bk( d ) =eχ ζd 0 k d|m0 a =1 (a0,d)=1 X Xd a0 φ(fχ) d −1 d 0 Bk( d ) = µ( )χ ( )χ(a )eχζfχ 0 φ(d) fχ fχ k fχ|d|m0 a =1 (a0,d)=1 0 X φ(f ) d d Xd B ( a ) = χ µ( )χ−1( ) χ(a0) k d e ζ χ fχ φ(d) fχ fχ 0 k fχ|d|m0 a =1 (a0,d)=1 X φ(f ) d d Y ¡ ¢ = χ µ( )χ−1( )d1−k (1 − χ(p)pk−1) −L(χ, 1 − k) e ζ . χ fχ φ(d) fχ fχ fχ|d|m0 p|d p-fχ ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 11 Now note that in this last sum only those d contribute for which d/fχ is both squarefree and prime to fχ. Hence this last expression becomes µ ¶ Y χ−1(p) ¡ ¢ = 1 − (p1−k − χ(p)) −f 1−kL(χ, 1 − k) e ζ φ(p) χ χ fχ p|m0 p-fχ µ ¶ Y p − χ−1(p)p1−k ¡ ¢ = −f 1−kL(χ, 1 − k) e ζ . p − 1 χ χ fχ p|m0 p-fχ On the other hand, using Lemma 1.1 again, we have X X φ(fχ) d −1 d (18) eχξm0 = eχζd = µ( )χ ( )eχζfχ φ(d) fχ fχ m1|d|m0 m1|d|m0 Y µ(p)χ−1(p) = e ζ φ(p) χ fχ p|m0 p-fχ Q since the only contributing summand is for d/f = p. Combining (17) χ p|m0,p-fχ and (18) we find Y k k−1 1 1−k ¡ 1−k ¢ eχ(D g)(0) = (1 − χ(l)l ) (p − χ(p)p) −fχ L(χ, 1 − k) eχξm0 m0 p|m0 p-fχ which together with (16) implies Y k 1 1−k k 1−k k−1 χχcyclo(λ) = − fχ (1 − χ(p)p )p (1 − χ(l)l )L(χ, 1 − k) m0 p|m0 p-fχ in Zl(χ). On the other hand by (4) combined with the Euler system relations [9][5.16] for the elements θd we have Y k # k # −1 χχcyclo(κ θm0 ) =χχcyclo(κ (1 − Frp )θfχ ) p|m0 p-fχ Y = (1 − χ(p)pk−1)(1 − χ(l)lk−1)L(χ, 1 − k). p|m0 p-fχ Hence 1 Y 1 − χ(p)pk χχk (λ) = − f 1−k p1−k χχk (κ#θ ) cyclo χ k−1 cyclo m0 m0 1 − χ(p)p p|m0 p-fχ The definition of vp for p | m0 ensures that Y Y k 1−k 1−k χχcyclo( vp) = fχ p p|m0 p|m0 p-fχ 12 MATTHIAS FLACH and the definition of ep ensures that k k χχcyclo(ep(1 − Frp)) = 1 − χ(p)p . k Since characters of the form χχcyclo are dense in Spec(Λ) we conclude à ! Y 1 ep(1 − Frp) # λ = − vp κ (θm ) Frp 0 m0 ep(1 − ) p|m0 p which finishes the proof of Proposition 2.2 for m0 > 1. For m0 = 1 the norm ∞ compatible system η1 does not lie in Uloc but only in the middle term of (7), and # 1 κ (θ1) only lies in γ−1 Λ, which is why we have to multiply with γ−1. The Coleman power series of (γ − 1)η1 is (1 + X)M − 1 f(X) = X with M := χcyclo(γ) = 1 + `m0. If we define g(X) as in (15) then f 0(X) Dkg(X) = (1 − φlk−1)Dk−1(1 + X) f(X) M(1 + X)M 1 + X = (1 − φlk−1)Dk−1 − (1 + X)M − 1 X µ ¶ d k−1 MeMz ez = (1 − φlk−1) − dz eMz − 1 ez − 1 µ ¶ d k−1 X∞ B (1) zn−1 = (1 − φlk−1) n (M n − 1) dz n (n − 1)! n=1 and B (1) (Dkg)(0) = (1 − lk−1) k (M k − 1) = − (M k − 1)(1 − lk−1)ζ(1 − k) k k # =χcyclo(−(γ − 1)κ (θ1)). This finishes the proof for m0 = 1. ¤ Lemma 2.3. For any integer d and Dirichlet character χ of conductor fχ | d one has 0 Xd B ( a ) Y χ(a0) k d = −d1−k (1 − χ(p)pk−1)L(χ, 1 − k) k a0=1 p|d 0 (a ,d)=1 p-fχ Proof. For d = fχ this is the well known formula for Dirichlet L-values in terms of Bernoulli polynomials [19][Thm.4.2] and the general case is proved by induction on the number of primes dividing d/fχ. The distribution relations for the Bernoulli ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 13 polynomials imply dp p−1 X a Xd X b + nd χ(a)B ( ) = χ(b) B ( ) k dp k dp a=1 b=1 n=0 (a,d)=1 (b,d)=1 Xd b = χ(b)p1−kB ( ) k d b=1 (b,d)=1 whereas on the other hand dp dp X a X a Xd b χ(a)B ( ) = χ(a)B ( ) + χ(bp)B ( ) k dp k dp k d a=1 a=1 b=1 (a,d)=1 (a,dp)=1 (b,d)=1 with the convention χ(p) = 0 if p | d. Hence dp X a Xd b χ(a)B ( ) = (p1−k − χ(p)) χ(b)B ( ). k dp k d a=1 b=1 (a,dp)=1 (b,d)=1 ¤ We now discuss briefly in what sense Proposition 2.2 can be regarded as a func- tional equation of the l-adic L-function L. Artin-Verdier duality gives an exact triangle of perfect complexes of Λ-modules 1 ∞ 1 ∞,∗ ∗ ∞ (19) RΓc(Z[ ],Tl ) → RΓc(Z[ ],Tl (1)) [−3] → C(Tl ) → m0l m0l where M ∞ ∼ ∞ ∞ C(Tl ) = RΓ(Qp,Tl ) ⊕ C(R,Tl ) p|m0l ∞ and C(R,Tl ) is quasi-isomorphic to a complex ∞ 1−φ∞ ∞ 1+φ∞ ∞ Tl −−−−→ Tl −−−−→ Tl with φ∞ ∈ Gal(Q¯ /Q) denoting any complex conjugation. This is shown by adapting the arguments in the proof of [6][Prop. 7.2], replacing the projective Al-module Tl ∞ ¯ by the projective Λ-module Tl . The isomorphism of Gal(Q/Q)-modules ∞,∗ ∼ ∞ Tl (1) ⊗ Λ = Tl Λ,κ# induces an isomorphism ∞,∗ 1 ∞,∗ ∼ 1 ∞ (20) ∆ [−3] ⊗ Λ = RΓc(Z[ ],Tl (1)) ⊗ Λ = RΓc(Z[ ],Tl ). Λ,κ# m0l Λ,κ# m0l Hence the exact triangle (19) can be rewritten ∞,∗ ∞ ∞ (21) ∆ [−3] ⊗ Λ → ∆ → C(Tl ) → . Λ,κ# For brevity we now set Q := Q(Λ) and denote the scalar extension to Q(Λ) by a subscript Q. 14 MATTHIAS FLACH ∞ Lemma 2.4. The complex C(R,Tl )Q has cohomology ∞ φ∞=1 ∼ ∞ ∼ ∞ (Tl,Q) = Y{v|∞},Q ⊗ Q = Y{v|lm ∞},Q ⊗ Q i = 0 Q,κ# 0 Q,κ# i ∞ H C(R,Tl )Q = (T ∞ )φ∞=−1 = Y ∞ =∼ Y ∞ i = 2 l,Q {v|∞},Q {v|lm0∞},Q 0 i 6= 0, 2 and 1 Det C(R,T ∞) = Λ · (2σ + σ0) ⊂ Det C(R,T ∞) . Λ l 2 Q l Q 0 0 where σ = σ ⊗1. Note here that eψσ = 0 (resp. eψσ = 0) is ψ is even (resp. odd). 0 Proof. Note that there is an isomorphism of Z [G n ]-modules H (Spec(L n ⊗ L l m0l m0l Q ¯ ∼ n Q), Zl(1)) = τ Zl where the sum is over the set of embeddings of Lm0l into C 0 n n and so we may naturally view Y{v|∞}(Lm0l ) as a quotient of H (Spec(Lm0l ⊗Q Q¯ ), Zl(1)) by the action of c − 1 where c ∈ Λ is the complex conjugation. Passing to the inverse limit over n we find that 2 ∞ ∞ ∞ ∞ ∞ ∞ H C(R,Tl ) = Tl /(φ∞ + 1)Tl = Tl /(c − 1)Tl = Y{v|∞} ∞ since φ∞ acts on Tl via −c. Moreover 0 ∞ ∞ φ∞=1 ∞ ∼ ∞ H C(R,Tl ) = (Tl ) = (c − 1)Tl = Y{v|∞} ⊗ Λ. Λ,κ# After localizing at Q we have Y ∞ ∼ Y ∞ {v|∞},Q = {v|m0l∞},Q since Y ∞ is finite free over Z and C(R,T ∞) is isomorphic to the complex of {v|m0l} l l Q Q-modules ∞,+ ∞,− (2,0) ∞,+ ∞,− (0,2) ∞,+ ∞,− Tl,Q ⊕ Tl,Q −−−→ Tl,Q ⊕ Tl,Q −−−→ Tl,Q ⊕ Tl,Q ∞ where ± denotes the ±-eigenspaces for c. The determination of DetΛ C(R,Tl ) ∞ inside DetQ C(R,Tl )Q is then an easy explicit computation (see also the very end of the proof of [6, Prop. 7.2]). ¤ ∞ Lemma 2.5. For any prime p 6= l the complex RΓ(Qp,Tl )Q is acyclic and one has e (1 − Frp ) ∞ p p ∼ ∞ DetΛ RΓ(Qp,Tl ) = Λ ⊂ Q = DetQ RΓ(Qp,Tl )Q. ep(1 − Frp) Proof. We follow the ideas in the proof of [6][Prop. 7.1]. Let R be either a commu- tative pro-l ring, or a localization of such a ring, and let C be a perfect complex of ¯ ∼ ˆ −1 R-modules with a continuous action of Gal(Fp/Fp) = Z · φp . Then the continuous cohomology complex 1−φ−1 ∼ p RΓ(Fp,C) = Tot(C −−−−→ C) is a perfect complex of R-modules where ‘Tot’ denotes the total complex of a double complex. The identity map of C induces an isomorphism ∼ idC,triv : R = DetR RΓ(Fp,C) ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 15 of (graded) invertible R-modules. If in addition the complex RΓ(Fp,C) is acyclic, −1 i.e. the differential dC = 1 − φp is a quasi-isomorphism, then there is a second isomorphism ∼ dC,triv : R = DetR RΓ(Fp,C) arising from acyclicity. Both isomorphisms are functorial for exact triangles in the variable C and commute with scalar extension. They are related by the formula [4][Lemma 1] −1 idC,triv = [1 − φp ]dC,triv −1 × where [1 − φp ] denotes the class in K1(R) = R . ∞ We apply these remarks to R = Q and C = RΓ(Ip,Tl,Q) where Ip temporarily denotes the inertia subgroup of Gal(Q¯ p/Qp). Then ∞ ∼ ∞ ∼ ∞ (22) RΓ(Fp,RΓ(Ip,Tl,Q)) = RΓ(Qp,Tl,Q) = RΓ(Qp,Tl )Q and there is an exact triangle of perfect complexes of Q-modules 0 ∞ ∞ 1 ∞ H (Ip,Tl,Q)[0] → RΓ(Ip,Tl,Q) → H (Ip,Tl,Q)[−1]. 0 ∞ ∞ Ip 1 ∞ ∼ ∞ On the Q-module H (Ip,Tl,Q) = (Tl,Q) (resp. H (Ip,Tl,Q) = (Tl,Q)Ip (−1)) the −1 × map 1 − φp acts by the element ep(1 − Frp /p) (resp. ep(1 − Frp)) of Q . By the discussion above we have id ∞ ⊗ Q = id ∞ RΓ(Ip,Tl ),triv Λ RΓ(Ip,Tl,Q),triv −1 = id ∞,Ip ⊗ id ∞ (T )I (−1),triv Tl,Q ,triv l,Q p −1 −1 =ep(1 − Frp /p)d ∞,Ip ⊗ ep(1 − Frp) d(T ∞ ) (−1),triv Tl,Q ,triv l,Q Ip Frp ep(1 − p ) = d ∞ . RΓ(Ip,Tl,Q),triv ep(1 − Frp) Fr e (1− p ) Together with (22) his means that the lattice p p Λ ⊆ Q is mapped to ep(1−Frp) ∞ ∞ DetΛ RΓ(Qp,Tl ) ⊂ DetQ RΓ(Qp,Tl,Q) under d ∞ , i.e. the statement of the Lemma. ¤ RΓ(Ip,Tl,Q),triv The following statement might be regarded as a functional equation of L. Proposition 2.6. Under the isomorphism ¡ ¢ O ∞ ∼ ∞ ∞ ∞ DetQ ∆Q = DetQ ∆Q ⊗ Q ⊗Q DetQ RΓ(Qp,Tl )Q ⊗ DetQ C(R,Tl )Q Q,κ# p|m0l induced by the exact triangle (21)Q we have à ! Y e (1 − Frp ) # p p −1 ¡ ∞ ¢−1 1 0 (23) L = κ (L) ⊗ −m0 vp · βm0 ⊗ (2σ + σ ) ep(1 − Frp) 2 p|m0 Proof. The long exact cohomology sequence of the triangle (21)Q reads 0 ∞ 2 ∞ ∗ 1 1 ∞ 0 → H C(R,Tl )Q → H (∆ )Q ⊗ Q → H (∆∞)Q → H (Ql,Tl )Q → Q,κ# 1 ∞ ∗ 2 ∞ 2 ∞ → H (∆ )Q ⊗ Q → H (∆ )Q → H C(R,Tl )Q → 0 Q,κ# 16 MATTHIAS FLACH i i ∞ or more concretely, given the computation of H (∆) (resp. H (Ql,Tl )) in section 1.2 (resp.(6)) (24) 0 → Y ∞ ⊗ Q → Y ∞,∗ ⊗ Q → U ∞ → U ∞ → {v|∞},Q {v|∞},Q {v|lm0},Q loc,Q Q,κ# Q,κ# ∞,∗ ∞ ∞ → U{v|lm },Q ⊗ Q → Y{v|∞},Q → Y{v|∞},Q → 0. 0 Q,κ# Q Noting that Q = Q(ψ) is a product ring (see (5)) this sequence becomes for even ψ 0 → 0 → 0 → U ∞ → U ∞ → {v|lm0},Q(ψ) loc,Q(ψ) ∞ ∞ → 0 → Y{v|∞},Q(ψ) → Y{v|∞},Q(ψ) → 0. Taking the bases of the modules in this sequence and using Proposition 2.2 we find à ! Y e (1 − Frp ) −1 p p −1 # −1 ∞ −1 2 · ηm0 ⊗ σ = −m0 vp κ (θm0 ) · (βm0 ) ⊗ 2 · σ ep(1 − Frp) p|m0 which is the even component of equation (23). For odd ψ the sequence (24) reads ∞ ∞,∗ ∞ 0 → Y{v|∞},Q(ωψ−1) → Y{v|∞},Q(ωψ−1) → 0 → Uloc,Q(ψ) → ∞,∗ → U −1 → 0 → 0 → 0 {v|lm0},Q(ωψ ) and a similar application of Proposition 2.2 gives the odd component of (23). ¤ Remark. Note that the element à ! Y e (1 − Frp ) p p −1 ¡ ∞ ¢−1 1 0 −m0 vp · βm0 ⊗ (2σ + σ ) ep(1 − Frp) 2 p|m0 is a Λ-basis of O ∞ ∞ DetΛ RΓ(Qp,Tl ) ⊗Λ DetΛ C(R,Tl ) p|m0l by Proposition 1.4, Lemma 2.5 and Lemma 2.4. Proposition 2.6 then implies that L ⊗ (κ#(L))−1 is a Λ-basis of ∞ ¡ ∞ ¢−1 DetΛ ∆ ⊗Λ DetΛ ∆ ⊗ Λ Λ,κ# which is also a (rather weak) consequence of Theorem 1.2. 3. The proof of Theorem 1.2 for l = 2 3.1. Recollections from [9]. In this section we continue the proof of Theorem 1.2 for l = 2 where we have left it off in [9]. By [9][Lemma 5.3] it suffices to show that ∞ (25) Λq ·L = DetΛq ∆q for all height one primes q of Λ, and primes q not containing l = 2 have already been dealt with in [9]. In the following l will always denote 2 and q a height one prime of Λ containing tor ∼ 2. Such primes are in bijection with Q -rational characters ψ of G ∞ G of l q m0l = m0` ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 17 order prime to l. We let c ∈ Λ be the complex conjugation. Then it was shown in ∞ [9][p.95] that ∆q is quasi-isomorphic to a complex 0 ∞,1 ∂ ∞,2 ∆q −→ ∆q ∞,i where both Λq-modules ∆q are free of rank one and the image of the differential 0 ∞,2 ∞,i ∂ is (c − 1)∆q . Following [9] we pick Λq-bases γi of ∆q so that 2 ∞ 1 ∞ γ2 7→ σ ∈ H (∆q ); (c + 1)γ1 = ηm0 ∈ H (∆q ) × and define α ∈ (c − 1)Λq by 0 ∂ (γ1) = αγ2. These conditions determine γ1, γ2 and therefore α up to factors in 1 + (c − 1)Λq. The class of α in 2 ∼ (c − 1)Λq/(c − 1) Λq = (c − 1)Λq/2(c − 1)Λq = Λq/(2, c + 1) is therefore independent of all choices and contains the same information as the Yoneda two-extension class in 2 2 ∞ 1 ∞ ∼ ExtΛq (H (∆q ),H (∆q )) = Λq/(2, c + 1) ∞ which is represented by the complex ∆q . We have ∞ −1 DetΛq ∆q = Λq · γ1 ⊗ γ2 = Λq · u ·L × for some u ∈ Q(Λ) and (25) becomes the statement that u ∈ Λq . We may compute ∼ Q u = (uψ) over Q(Λ) = ψ Q(ψ), a product of fields indexed by the Ql-rational characters Gm0`. For even ψ (with prime-to-l part ψq) we have as in [9] µ ¶ ¡ ¢ 1 e γ−1 ⊗ γ = e ( η )−1 ⊗ σ = e L ψ 1 2 ψ 2 m0 ψ ∞ and hence uψ = 1. For odd ψ the canonical basis of ∆ ⊗Λ Q(ψ) arising from the −1 acyclicity of this complex is γ1 ⊗ αγ2 and hence µ ¶ µ ¶ ¡ ¢ 1 θ e γ−1 ⊗ γ = e · γ−1 ⊗ αγ = e m0 ·L ψ 1 2 ψ α 1 2 ψ α ³ ´ θm0 so that uψ = eψ α . In [9] we were erroneously assuming that α = 1 − c and × that the resulting u lies in Λq . We shall show in the following that α = θm0 (1 + (c − 1)x + (c + 1)y) for some × x, y ∈ Λq. This then implies that u = 1 + (c − 1)x, i.e. u ∈ Λq . The complex 1 ∞ ∞ RΓc(Z[ ],T ) turns out to be more amenable to direct analysis than ∆ (recall m0l l that the two are related by (20)). After describing an explicit representative of this complex and a basis of its determinant we continue the computation of α in section 3.4. 1 ∞ 3.2. Explicit construction of RΓc(Z[ ],T ). For a profinite group G and a m0l l continuous G-module M we denote by RΓ(G, M) (resp. Zi(G, M)) the complex of continuous cochains (resp. the group of continuous cocycles of degree i). We let Σ be the set of primes dividing m0l, LΣ the maximal algebraic extension of Q 18 MATTHIAS FLACH unramified outside Σ and set GΣ = Gal(LΣ/Q). There is a natural isomorphism [16][Prop. 2.9] 1 ∞ ∼ ∞ RΓ(Z[ ],Tl ) =RΓ(GΣ,Tl ) m0l ∞ ∞ and we shall also identify RΓ(Qp,Tl ) and RΓ(R,Tl ) with the complex of contin- uous cochains of the respective Galois group. By definition M 1 ∞ ∞ ∞ (26) RΓc(Z[ ],Tl ) = Cone RΓ(GΣ,Tl ) → RΓ(Qp,Tl ) [−1]. m0l p|m0l∞ We denote by 0 ∞ x7→(g7→(g−1)x) 1 ∞ 1 d := d : Tl −−−−−−−−−−→ Z (GΣ,Tl ) = ker(d ) ∞ the 0-th differential in the complex RΓ(GΣ,Tl ) and by resp (resp. resp) the re- striction map of cochains (resp. cohomology classes) to a fixed decomposition group at p. 1 ∞ Proposition 3.1. a) The complex RΓc(Z[ ],T )q is naturally quasi-isomorphic m0l l to the complex concentrated in degrees 1 and 2 ∂ 1 ∞ (27) Kq −→ H (Ql,Tl )q where 1 ∞ K := {z ∈ Z (GΣ,Tl ) | res∞(z) = 0} and ∂(z) := reslz. b)The natural map ∞ 0 ∞ ∼ 1 1 ∞ ∼ (c − 1)Tl,q = H (R,Tl )q = Hc (Z[ ],Tl )q = ker(∂), m0l where the second isomorphism arises from the cohomology sequence of (26), is given by x 7→ dx, and the map 1 ∞ ∼ 2 1 ∞ H (Ql,Tl )q → coker(∂) = Hc (Z[ ],Tl )q m0l is the one in the cohomology sequence of (26). 1 ∞ c) The modules Kq and H (Ql,Tl )q are both free of rank one over Λq. Proof. By Lemma 3.2 below we may ignore the summands for primes p 6= l, ∞ in ∞ (26) and for p = l (resp. p = ∞) the complex RΓ(Qp,Tl )q is concentrated in degree 1 (resp. 0). For p = l this follows from (6) and for p = ∞ the complex ∞ RΓ(R,Tl ) is quasi-isomorphic to ∞ c+1 ∞ c−1 ∞ c+1 Tl −−→ Tl −−→ Tl −−→ .... i ∞ Moreover, a computation using Artin-Verdier duality shows that H (GΣ,Tl )q = 0 1 ∞ for i ≥ 2 and hence that RΓc(Z[ ],T )q is naturally quasi-isomorphic to m0l l ¡ ≤1 ∞ 1 ∞ ≤1 ∞ ¢ Cone τ RΓ(GΣ,Tl ) → H (Ql,Tl )[−1] ⊕ τ RΓ(R,Tl ) q [−1]. ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 19 Written out explicitly this mapping cone is the q-localization of the complex in the first row of the following commutative diagram ∞ a ∞ 1 ∞ b 1 ∞ 1 ∞ T −−−−→ T ⊕ Z (GΣ,T ) −−−−→ Z (R,T ) ⊕ H (Ql,T ) xl l x l l x l (28) i i z7→reslz 1 ∞ 0 −−−−→ K −−−−−→ H (Ql,Tl ) where a(x) = (x, −dx), b(x, y) = (res∞(dx + y), resly) and i(z) = (0, z). We show that the vertical map of complexes is a quasi-isomorphism. Since a is injective we get a cohomology isomorphism in degree 0. If (x, y) ∈ ker(b)/ im(a) then (29) (x, y) = (0, dx + y) + (x, −dx) ≡ i(dx + y) mod im(a) and dx + y ∈ ker(∂) since resl ◦ d = 0. If z ∈ ker(∂) and i(z) = (0, z) = (x, −dx) ∈ im(a) then x = dx = z = 0, hence we get a cohomology isomorphism in degree ∞ res∞ ◦d 1 ∞ 1 ∞ 1. Since Tl −−−−−→ Z (R,Tl ) is surjective, for any given (ξ, η) ∈ Z (R,Tl ) ⊕ 1 ∞ ∞ H (Ql,Tl ) we find x ∈ Tl so that (ξ, η) = (0, η) + (res∞(dx), 0) ≡ i(η) mod im(b). On the other hand if i(η) = (0, η) = (res∞(dx + y), resly) ∈ im(b) then dx + y ∈ K and η = resl(dx + y) ∈ im(∂). Hence we get a cohomology isomorphism in degree 2 ≤1 ∞ 1 ∞ and the statement a). The map of complexes τ RΓ(R,T ) → RΓc(Z[ ],T )[1] l m0l l is explicitly given by T ∞ −−−−−→res∞ ◦d Z1(R,T ∞) l l (30) y y ∞ a ∞ 1 ∞ b 1 ∞ 1 ∞ Tl −−−−→ Tl ⊕ Z (GΣ,Tl ) −−−−→ Z (R,Tl ) ⊕ H (Ql,Tl ) where the vertical maps are the inclusions into the respective first summands. Eval- uating a 1-cocycle f on a complex conjugation φ∞ ∈ GΣ gives an isomorphism 1 ∞ ∼ ∞, φ∞=−1 ∼ ∞, c=1 ∞ (31) Z (R,Tl ) = Tl = Tl = (c + 1)Tl ∞ and the differential res∞ ◦d is given by x 7→ (c + 1)x. Its kernel is (c − 1)Tl which 1 ∞ 1 ∞ maps into Z (R,Tl )⊕H (Ql,Tl ) via x 7→ (x, 0). The first part of b) then follows from (29), and the second part is clear from the mapping cone construction. 1 ∞ ∞ The Λq-module H (Ql,Tl )q is free of rank one with basis βm0 by (6), (7), (8) and Lemma 1.3. A diagram chase using (31) shows that the following commutative 20 MATTHIAS FLACH diagram has exact rows and columns (even before localization at q) (32) 0 0 0 y y y ∞ π 1 ∞ 0 −−−−→ (c − 1)T −−−−→ K −−−−→ H (GΣ,T ) −−−−→ 0 l l y y k ∞ d 1 ∞ 1 ∞ 0 −−−−→ T −−−−→ Z (GΣ,T ) −−−−→ H (GΣ,T ) −−−−→ 0 l l l y res∞y y 0 −−−−→ T ∞/(c − 1)T ∞ −−−−→∼ Z1(R,T ∞) −−−−→ 0 −−−−→ 0 l l l y y y 0 0 0 Combining this diagram with the isomorphism [9][p.94] à ! 1 ∞ ∼ 1 × ∼ 1 ∞ H (G ,T ) = lim O n [ ] ⊗ Z = H (∆ ) = (Λ /(c − 1)) · η Σ l q Lm0l Z l q q m0 ←− m0l n q we find that Kq is isomorphic to an extension of Λq/(c − 1) by Λq/(c + 1). On the 1 ∞ other hand, Kq is of finite projective dimension over Λq since RΓc(Z[ ],T )q is m0l l perfect. An argument as in [9][eq.(5.17)] then shows that Kq must be isomorphic to Λq. ¤ ∞ Lemma 3.2. For p 6= l, ∞ the complex RΓ(Qp,Tl )q is acyclic. ∞ Proof. Similarly to (6) the cohomology of RΓ(Qp,Tl ) is naturally isomorphic to × n Q (lim Q(ζ n ) /l ) ⊗ Z =∼ Z (1) i = 1 ←−n m0l p Zl l l v|p i ∞ ∼ Q H (Qp,Tl ) = Zl i = 2 v|p 0 otherwise n n where the limit is taken with respect to the norm maps and Q(ζm0l )p = Q(ζm0l )⊗Q Qp denotes a finite product of local fields. Since the µ-invariant of any Zl[[γ − 1]]- module which is finite free as a Zl-module vanishes we conclude by [9][Lemma 5.6] i ∞ that H (Qp,Tl )q = 0. ¤ 3.3. Description of a basis of Kq. In order to describe elements in Kq we shall use the computations in the appendix to construct a set-theoretic section z of the map π in (32). Lemma 3.3. Let ∞ 1 × ∼ 1 ∞ u = (un) ∈ U := lim OL n [ ] ⊗Z Zl = H (GΣ,T ) {v|m0l} ←− m0l l n m0l be a norm compatible system of real units, i.e. so that (c − 1)u = 0. Then if z ∈ K is any lift (i.e. π(z) = u) and we write (c − 1)z = d(λ∞σ) ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 21 then λ∞ = (λn) ∈ (c − 1)Λ satisfies X −1 λ ≡ ² τ n mod 2 n n,a a,m0l n a mod m0l n ² τa,m0l n,a n where ²n,a ∈ {0, 1} is defined by (−1) = sgn(σm0l (un )). The class of λ∞ modulo 2(c − 1)Λ only depends on u. Proof. Since 0 = (c + 1)(c − 1)z = (c + 1)d(λ∞σ) = d((c + 1)λ∞σ) and d is injective we have (c + 1)λ∞ = 0, i.e. λ∞ ∈ (c − 1)Λ. Changing z to ∞ z + d((c − 1)x) with x ∈ Tl , see diagram (32), will clearly change λ∞ by an 2 element in (c − 1) Λ = 2(c − 1)Λ. We conclude that u determines λ∞ modulo 2(c − 1)Λ. n n In order to prove the congruence for λn ∈ Λn := Z/l Z[Gm0l ] we apply the discussion in the appendix to G = Γn, H = Hn and M = µln where n n n Hn = Gal(Fm0l /Lm0l ), Γn = Gal(Fm0l /Q) n n and Fm0l denotes the maximal abelian extension of Lm0l unramified outside Σ n n n and annihilated by l . We denote by σm0l : Fm0l → C an embedding which n n n extends σm0l on Lm0l and note that the embedding σm0l determines a surjection Gal(Q¯ /Q) → Γn which allows us to view φ∞ as an element of Γn. We let S = Sn be a set of representatives of Hn\Γn which not only contains 1 but is also right invariant under φ∞. This we can achieve by picking a set S˜ ⊂ Γn ∼ n which maps to a set of right coset representatives of φ∞ ∈ G/H = Gm0l and setting S = S˜ ∪ Sφ˜ ∞. Then we have h(x) = h(xφ∞) and the cocycle ι(f) constructed in Lemma 4.1 therefore satisfies (33) ι(f)(φ∞) = 0 for any homomorphism f : Hn → µln . We have an isomorphism of free, rank one Λn-modules Γn Coind µ n = Hom (Z[Γ ], µ n ) ∼ Hom n (Λ , µ n ) ∼ Hn l Z[Hn] n l = Z/l Z n l = 0 n ¯ n n H (Spec(Lm0l ⊗Q Q), µl ) = Map(Tn, µl ) =: Fn n ¯ n ¯ where Tn = HomRings(Lm0l , Q) with its natural action of Gm0l and Gal(Q/Q) and X n n ˆ n (34) HomZ/l Z(Λn, µl ) 3 ψ 7→ ψ(g)g · σm0l . n g∈Gm0l n g−1 ψˆ(g) ˆ n Here ψ(g) ∈ Z/l Z is defined by ψ(g) = ζln and we also denote by σm0l the n n element of Fn sending σm0l to ζl and all other τ ∈ Tn to 0. Moreover 1 1 1 1 ∞ lim H (H , µ n ) ∼ lim H (Γ , F ) ∼ lim H (G , F ) ∼ H (G ,T ) ←− n l = ←− n n = ←− Σ n = Σ l n n n where the first isomorphism is given by the Shapiro Lemma. The norm compatibility of the un implies that the homomorphisms √ ln h−1 1 (35) fn(h) := ( un) ∈ H (Hn, µln ) 22 MATTHIAS FLACH are compatible under the corestriction maps, and equation (33) implies that (36) z(u) := (ι(f )) ∈ lim Z1(Γ , F ) ∼ Z1(G ,T ∞) n n ←− n n = Σ l n does in fact lie in K. Note that the elements ι(fn) are norm-compatible by Lemma 0 0 n n−1 n−1 4.2 applied to H = Gal(Fm0l /Lm0l ) and M = µl , and that there is an iso- 0 0 ∼ 0 morphism of Γn-modules HomZ[Hn−1](Z[Γn−1],M ) = HomZ[H ](Z[Γn],M ). From now on we denote by z(u) this particular lift of u (depending on a choice of sets of representatives Sn formed for increasing n as in Lemma 4.2) and define λ∞ by (c − 1)z(u) = d(λ∞σ). n Let ψn ∈ HomZ[Hn](Z[Γn], µl ) be the element associated to fn by Lemma 4.3. Since sφ∞ ∈ Sn for any s ∈ Sn we have φ∞s = hφ∞,ssφ∞ and hence by Lemma 4.3 √n −1 √n −1 l φ∞s(x)φ∞s(x) −1 l s(x)φ∞s(x) −1 ψn(x) = cfn(hc,x) = c( un) = ( un) −1 −1 for x ∈ G/H = G n . Now take x := τ = τ n . Since we are only interested m0l a a,m0l n−1 in the value of ψn ”modulo 2”, i.e. in its l -st power, it suffices to compute the −1 n action of s(x)φ∞s(x) on a fixed square root wn ∈ Fm0l of un. The element −1 s(x) τa wn is a square root of un and −1 −1 φ∞s(x) s(x) wn = ±wn τ n a according to whether σm0l (un ) is positive or negative. Hence −1 s(x)φ∞s(x) ²a,n wn = (−1) wn n−1 −1 l ²a,n 0 and ψ (τ n ) = (−1) . Writing ψ = λ σ n as in (34) we have n a,m0l n n m0l X X 0 ˆ −1 (37) λ = ψ (g)g ≡ ² τ n mod 2. n n a,n a,m0l n n g∈Gm0l a mod m0l We did not verify that the ψn are norm-compatible, so we do not know a priori that 0 n λn = λn (note that λnσm0l is the unique norm compatible choice of elements as in Lemma 4.3). However, using the norm-compatibility of un, the explicit definition of 0 0 ¯ n ²a,n and (37) one immediately verifies that the images λn of λn in Z/2Z[Gm0l ] are ¯0 ¯ norm compatible. Hence δn := λn − λn is normP compatible and, since d(δn) = 0, for any n we have δn ∈ {0,Nn} where Nn = g. Since the norm of Nn is g∈G n m0l zero for n ≥ 2 the only norm compatible choice is δn = 0 and we conclude that λn satisfies the required congruence. ¤ The following proposition is the heart of the proof of Theorem 1.2. Proposition 3.4. There exists a Λq-basis zm0 of Kq so that ∞ (c − 1)zm0 ≡ d((γ − χcyclo(γ))θm0 σ) mod 2(c − 1)d(Tl ) and # ∞ ∂(zm0 ) = ν · (γ − χcyclo(γ))κ (θm0 ) · βm0 ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 23 where à ! Y 1 ep(1 − Frp) (38) ν = − vp . Frp m0 ep(1 − ) p|m0 p Proof. We define à ! M 1 − ζ n γ−χcyclo(γ) m0l zm0 := z(ηm0 ) = z M (1 − ζm ln ) 0 n where M = χcyclo(γ) = 1 + 4m0 and z(u) was defined in (36). Note here that 1 ∞ γ −χcyclo(γ) is a unit in Λq and that ηm0 is a Λq-generator of H (GΣ,Tl )q so that zm0 is indeed a basis of Kq by Nakayama’s Lemma. Then ∂(zm0 ) = resl(zm0 ) is given by Proposition 2.2. Note that for m0 = 1 × the factor γ − 1 in Prop. 2.2 can be canceled since γ − 1 ∈ Λq . Defining λ∞ by (c − 1)zm0 = d(λ∞σ) we deduce from Lemma 3.3 and Lemma 3.5 below ¹ º Ma λn,a ≡ n mod 2. m0l On the other hand X µ ¶ X µ ¶ a 1 −1 Ma M −1 n (γ − χcyclo(γ))gm0l = − n − γτa + n − τa m0l 2 m0l 2 a µ¿ À ¶ a µ ¶ X Ma 1 X Ma M = − − τ −1 + − τ −1 m ln 2 a m ln 2 a a 0 a 0 X µ¹ º ¶ Ma 1 − M −1 = n + τa m0l 2 a ¹ º X Ma ≡ τ −1 mod 2 m ln a a 0 n where the sums are over 0 < a < m0l ,(a, lm0) = 1 and < x >= x − bxc. Hence à ! X −1 n (γ − χcyclo(γ))θm0 = ((γ − χcyclo(γ))gm0l )n ≡ λn,aτa = λ∞ mod 2. a n ¤ Lemma 3.5. Let M ≡ 1 mod 4 be an integer, 0 < x < 1 a real number and 1−uM bMxc u = exp(2πix). Then the sign of the real number (1−u)M is (−1) . Proof. For 0 < x < 1 the zeros of the real valued differentiable function f(x) = 1−uM 1 2 M−1 (1−u)M are at x = M , M ,..., M and all of these are simple. Hence f changes bMxc 1 sign precisely at those arguments and so does (−1) . For x = 2 we have M bMxc M−1 f(x) = 2/2 > 0 and (−1) = (−1) 2 = 1, so the signs of the two functions agree. ¤ 24 MATTHIAS FLACH ∞ 1 ∞ 3.4. Passing from ∆ to RΓc(Z[ ],T ). Recall that the two complexes are m0l l related by ∞,∗ ∼ 1 ∞ ∆ [−3] ⊗ Λ = RΓc(Z[ ],Tl ). Λ,κ# m0l ∗ # where P = R HomΛ(P, Λ) denotes the Λ-dual and κ :Λ → Λ is the automorphism −1 ∗ ∗# g 7→ χcyclo(g)g . For i = 1, 2 we denote by γi the dual basis to γi and by γi the ∗ ∞,i,∗ image of γi in ∆ ⊗ Λ. By Proposition 3.1 we may pick isomorphisms Λ,κ# ∞,2,∗ ∼ ∞,1,∗ ∼ 1 ∞ (∆ ⊗ Λ)q = Kq, (∆ ⊗ Λ)q = H (Ql,Tl )q Λ,κ# Λ,κ# ∗# and the following Lemma summarizes the information we have about γ2 from the defining properties of γ1, γ2 and α. Proposition 3.6. We have ∗# (39) (c − 1)γ2 = (c − 1)σ. and µ ¶ θ −1 (40) ∂(γ∗#) = κ#(α)κ#(ν%)−1 · m0 · β∞ 2 1 − c m0 θm0 × where ν (resp. %) was defined in (38) (resp. (12)) and 1−c denotes any x ∈ Λq satisfying (1 − c)x = θm0 . ∞,1 ∼ 1 ∞ Proof. We may pick a Λq-isomorphism ∆q = H (Ql,Tl )q so that the diagram ∞,1 ∼ 1 ∞ ∆q −−−−→ H (Ql,T )q x x l 1 ∞ ∼ 1 1 ∞ H (∆ ) −−−−→ H (Z[ ],T )q q m0l l commutes. If we set # κ (θm0 ) ∞ γ1 := ν · · β 1 + c m0 where ν was defined in (38) then γ1 satisfies the defining property (1 + c)γ1 = ηm0 by Proposition 2.2. Hence µ ¶−1 κ#(θ ) γ∗ = ν−1 · m0 · β∞,∗ 1 1 + c m0 and µ ¶ µ ¶ θ −1 θ −1 γ∗# = κ#(ν)−1 · m0 · β∞,∗# = κ#(ν)−1 · m0 · κ#(%)−1 · β∞ 1 1 − c m0 1 − c m0 ∞ # −1 ∞ 0 since < βm0 , κ (%) βm0 >= 1 by the definition of % in (12). Now ∂ (γ1) = αγ2 ∗ ∗ implies ∂(γ2 ) = αγ1 and µ ¶ θ −1 ∂(γ∗#) = κ#(α)γ∗# = κ#(α)κ#(ν%)−1 · m0 · β∞ 2 1 1 − c m0 ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 25 noting that ∂0 and ∂ are dual maps. This finishes the proof of (40). Concerning (39) we have a commutative diagram T ∞ × T ∞,∗(1) −−−−→ Λ l l x py i k H2(∆∞) × H1(Z[ 1 ],T ∞,∗(1)) −−−−→ Λ c m0l l where i is the inclusion 1 1 ∞,∗ ∼ 0 ∞,∗ ∞,∗ φ∞=1 ∞,∗ Hc (Z[ ],Tl (1)) = H (R,Tl (1)) = Tl (1) → Tl (1). m0l ∗ The defining property of γ2 that p(γ2) = σ translates into (c + 1)γ2 = (c + 1)σ and ∗# (c − 1)γ2 = (c − 1)σ. ¤ Denote by J ⊆ qΛq the Λq-ideal J := (c + 1, c − 1) = (2, c − 1) = (2, c + 1). By Proposition 3.4 the basis µ ¶ θ −1 z := (γ − χ (γ))−1 m0 z cyclo c − 1 m0 of Kq satisfies ∗# ∞ (c − 1)z ≡ (c − 1)σ = (c − 1)γ2 mod J(c − 1)Tl , hence ∗# z ≡ γ2 mod JKq. ∗# Since z is a basis of Kq we can write z(1 + j) = γ2 with j ∈ J. By Proposition 3.4 we have µ ¶−1 θm0 # ∞ ∂(z) = ν · · κ (θm ) · β c − 1 0 m0 ∗# and comparing this with ∂(γ2 ) in Proposition 3.6 we find # # # κ (α) = −κ (ν%) · ν · κ (θm0 ) · (1 + j) or, equivalently, # # α = − ν · % · κ (ν) · θm0 · (1 + κ (j)) # = θm0 · κ (ν) · ν · % · (1 + j1) # with j1 = −2 − κ (j) ∈ J. Recall that we need to show that α = θm0 (1 + j2) for some j2 ∈ J. This is then accomplished by the following Lemma. Lemma 3.7. For ν defined in (38) and % defined in (12) we have κ#(ν) · ν · % ∈ 1 + J. Proof. First note that X X # 1 # 1 −1 κ (eIp ) = κ (g) = g = eIp |Ip| |Ip| g∈Ip g∈Ip and therefore # # # # κ ep(x) = κ (1 + (x − 1)eIp ) = 1 + (κ (x) − 1)eIp = ep(κ (x)). 26 MATTHIAS FLACH Moreover # −1 κ (Frp) = χcyclo(Frp) Frp = p/ Frp and à ! −1 −p/ Fr (1 − Frp ) 1 − Frp # 1 − Frp 1 − p Frp p p p κ Fr = −1 = −1 = p . p 1 − Fr − Fr (1 − Fr ) 1 − Frp 1 − p p p p Hence Y # 1 # κ (ν) · ν · % = 2 % ep(p)κ (vp)vp. m0 p|m0 −2 Since m0 is an odd integerQ we have m0 ∈ 1 + J and it suffices to show that κ#(v )v ∈ 1+J and that % e (p) ∈ 1+J. Now κ#(v )v ∈ Λ× is associated p p p|m0 p p p via Lemma 2.1 to the function max(x,1) max(x,1) u : x 7→ (p/ Fr˜ p · Fr˜ p) = p , that is X X # κ (vp)vp = u(ordp(fχ))eχ = 1 + 2 v(ordp(fχ))eχ ∈ 1 + J χ χ where v is the function x 7→ (pmax(x,1) − 1)/2. Note here that v(0) − v(1) = (p − 1)/2 − (p − 1)/2 = 0 is indeed divisible by (p − 1) in Λ, so Lemma 2.1 applies. As to Y Y % ep(p) = cl%0 ep(p) p|m0 p|m0 we note that in the sum (13) the term φ(m0)/φ(d/(d, g + 1)) is odd precisely Qwhen all primes dividing m0 also divide d/(d, g + 1), i.e. precisely when d = pordp(g+1)+1 (and d | m ). So modulo 2 the sum (13) reduces to one nonzero p|m0 0 ordp(m0) term if g + 1 6≡ 0 mod p for all p | m0, and has all terms equal to zero if ordp(m0) there is a p | m0 with g + 1 ≡ 0 mod p . This implies X Y X Y −1 ¡ ¢ ¡ ¢ %0 = %0(g) · g ≡ g ≡ cp +e ˜Ip mod 2 g∈Gm0 p|m0 g∈Ip p|m0 g6=cp P where c is the projection of c to I ande ˜ = |I |e = g. Since e (p) = p p Ip p Ip g∈Ip p 1 + (p − 1)eIp ≡ 1 +e ˜Ip mod 2 we deduce Y Y Y ¡ ¢ % ep(p) ≡ cl (cp +e ˜Ip )(1 +e ˜Ip ) = cl cp + 2˜eIp + |Ip|e˜Ip p|m0 p|m0 p|m0 Y ≡ cl cp = c = 1 + (c − 1) ≡ 1 mod J. p|m0 This finishes the proof of the Lemma. ¤ ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 27 4. Appendix: Generalities on the Shapiro Lemma In this section we fix a group G with subgroup H and a set S ⊆ G of right coset S. representatives so that G = s∈SHs. Modules are understood to be left modules. We assume that S contains the identity and define h(g) ∈ H and s(g) ∈ S for g ∈ G by g = h(g)s(g). Then for h ∈ H and g ∈ G (41) h(hg) = h h(g), s(hg) = s(g). If H is normal in G we define for g1, g2 ∈ G ³ ´−1 s(g1) −1 (42) hg1,g2 := h(g2) h(g1) h(g1g2) ∈ H where ab = aba−1. Then one verifies that s(g1)s(g2) = hg1,g2 s(g1g2) which together with (41) implies that hg1,g2 only depends on the classes of g1, g2 in G/H. The function hg1,g2 satisfies s(g1) (43) hg1,g2 hg1g2,g3 = hg2,g3 hg1,g2g3 which is the standard 2-cocycle relation if H is abelian. For any H-module M we let G CoindH M := HomZ[H](Z[G],M) be the coinduced module with its natural left G-action given by (44) (gψ)(x) = ψ(xg). We denote by Z[G•] → Z the standard resolution, and similarly for H. Consider the Shapiro-Lemma quasi-isomorphism • ∼ • ρ (45) HomZ[G](Z[G ], HomZ[H](Z[G],M)) = HomZ[H](Z[G ],M) −→ • → HomZ[H](Z[H ],M) where the first isomorphism is the canonical adjunction and ρ is given by restriction of cochains to H. The map ρ has a section ι induced by the homomorphism n n Z[G ] → Z[H ], (g1, ..., gn) 7→ (h(g1), ..., h(gn)) which is H-equivariant by (41) and which depends on the choice of S. Lemma 4.1. The map ι induces a map 1 1 G 1 ι : C (H,M) → C (G, CoindH M) = C (G, HomZ[H](Z[G],M)) on inhomogeneous 1-cochains which for g, x ∈ G is given by −1 s(x) (ι(f)(g))(x) = h(x)f(h(x) h(xg)) = h(x)f( h(g)hx,g). This last expression holds if H is normal in G. 28 MATTHIAS FLACH Proof. To the inhomogeneous 1-cochain f with values in M there is associated, respectively, a homogenous cochain on H, its image under ι on G, its image un- der the adjunction with values in HomZ[H](Z[G],M) and its image by passing to inhomogeneous cochains on G as follows −1 (h1, h2) 7→ h1f(h1 h2) −1 (g1, g2) 7→ h(g1)f(h(g1) h(g2)) −1 (g1, g2) 7→ (x 7→ h(xg1)f(h(xg1) h(xg2)) g 7→ (x 7→ h(x)f(h(x)−1h(xg)). The second identity follow from (42). ¤ Now let H ⊆ H0 ⊆ G be a larger subgroup so that [H0 : H] < ∞, S0 (resp. E) a set of representatives of H0\G (resp. H\H0) and S = ES0. Denote by h0, s0 and ι0 the maps h, s and ι formed with respect to the set S0. P −1 0 Lemma 4.2. Let trH0/H = t∈E t be the trace operator on any left H -module, let M 0 be a H0-module with trivial action and let H1(H,M 0) = Hom(H,M 0) = Z1(H,M 0) ⊆ C1(H,M 0) be the group of one-cocycles. Then there is a commutative diagram 1 0 ι 1 0 Z (H,M ) −−−−→ C (G, HomZ[H](Z[G],M )) 0 H tr 0 corH y H /H y 0 1 0 0 ι 1 0 Z (H ,M ) −−−−→ C (G, HomZ[H0](Z[G],M )) Proof. The first isomorphism in (45) sits in a commutative diagram • 0 ∼ • 0 HomZ[G](Z[G ], HomZ[H](Z[G],M )) −−−−→ HomZ[H](Z[G ],M ) trH0/H y trH0/H y • 0 ∼ • 0 HomZ[G](Z[G ], HomZ[H0](Z[G],M )) −−−−→ HomZ[H0](Z[G ],M ) H H0 0 which arises from the trace map trH0/H : N → N applied to the H -module • 0 ∼ • 0 HomZ[G](Z[G ], HomZ(Z[G],M )) −→ HomZ(Z[G ],M ). There is a commutative diagram • 0 ι2 0• 0 ι1 • 0 HomZ[H](Z[G ],M ) ←−−−− HomZ[H](Z[H ],M ) ←−−−− HomZ[H](Z[H ],M ) trH0/H y trH0/H y 0 • 0 ι 0• 0 HomZ[H0](Z[G ],M ) ←−−−− HomZ[H0](Z[H ],M ) 0 n 0n where the maps ι and ι2 are induced by Z[G ] → Z[H ] sending (g1, ..., gn) to 0 0 0 (h (g1), ..., h (gn)). The commutativity follows from (41): For t ∈ E ⊆ H we −1 0 0 −1 0 0 have t φ(th (g1), ..., th (gn)) = t φ(h (tg1), ..., h (tgn)). Finally, the corestriction 1 0• 0 map on H (H,M) is induced by the trace map trH0/H on HomZ[H](Z[H ],M ) 0n n [2][Ch.III.9 (C)]. The quasi-isomorphism ι1 is induced by Z[H ] → Z[H ] sending 0 (g1, ..., gn) to (h(g1), ..., h(gn)) and we have ι = ι2 ◦ ι1 since h(h (g)) = h(g) (this follows from S = ES0). ¤ ON THE CYCLOTOMIC MAIN CONJECTURE FOR THE PRIME 2 29 We now assume that H is normal in G with quotient Q = G/H and that M is a G-module. Then there is a natural left Q-action on the coinduced module G G CoindH ResH M = HomZ[H](Z[G],M) given by (46) (qψ)(x) =qψ ˜ (˜q−1x) whereq ˜ ∈ G is any lift. This action commutes with the left G-action (44) and leads i G ∼ i to the natural Z[Q]-module structure on H (G, CoindH M) = H (H,M). The first • isomorphism in (45) is Z[Q]-equivariant where q ∈ Q acts on HomZ[H](Z[G ],M) via −1 −1 (qf)(x1, ..., xn) :=qf ˜ (˜q x1, ..., q˜ xn). The restriction map ρ and its section ι are only Z[Q]-equivariant up to homotopy • where HomZ[H](Z[H ],M) is given its natural Q-action −1 −1 (qf)(x1, ..., xn) :=qf ˜ (˜q x1q,˜ ..., q˜ xnq˜). Lemma 4.3. Assume that H E G and that M is a G-module with trivial H-action. Let f ∈ H1(H,M) = Hom(H,M) = Z1(H,M) be a cohomology class fixed by c ∈ Q = G/H. Then (c − 1)ι(f) = d(ψ) G where ψ ∈ CoindH M := HomZ[H](Z[G],M) is given for x ∈ G by ψ(x) := cf(hc−1,x) and G 1 G d : CoindH M → C (G, CoindH M) is the differential in the inhomogeneous standard complex. Proof. The fact that f is fixed by c translates into the identity f(s(c)h) = cf(h) (and the same for c replaced by c−1 and we could have chosen any representative for c in G in place of s(c)). From Lemma 4.1 and the fact that H acts trivially on M one has ¡ ¢ s(c−1x) s(x) (c − 1)ι(f)(g) (x) = cf( h(g)hc−1x,g) − f( h(g)hx,g) s(c−1x) s(x) = cf( h(g)) − f( h(g)) + cf(hc−1x,g) − f(hx,g) ks(c−1)s(x) s(x) = cf( h(g)) − f( h(g)) + cf(hc−1x,g) − f(hx,g) = cf(hc−1x,g) − f(hx,g)(47) −1 k where k = hc−1,x ∈ H so that f( h) = h for all h ∈ H. On the other hand, applying −1 f to (43) with g1 = c , g2 = x, g3 = g we find s(c−1) f(hc−1,x) + f(hc−1x,g) = f( hx,g) + f(hc−1,xg) −1 = c f(hx,g) + f(hc−1,xg) or equivalently −1 f(hc−1x,g) − c f(hx,g) = f(hc−1,xg) − f(hc−1,x). Multiplying with c and combining with (47) we get ¡ ¢ (c − 1)ι(f)(g) (x) = cf(hc−1,xg) − cf(hc−1,x) = (dψ)(x). ¤ 30 MATTHIAS FLACH References [1] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, In: The Grothendieck Festschrift I, 1990, pp. 333–400. [2] K.S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer, New York, 1982. [3] D. 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