ASPECTS OF IWASAWA THEORY OVER FUNCTION FIELDS 1 ...

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IWASAWA THEORY OVER FUNCTION FIELDS 24Theorem 6.2. The evaluation map ev : f ↦→ {f(ω n )} gives an isomorphism(A p ((x)) ∗ ) N =id ≃ lim←K ∗ nwhere the inverse limit is taken with respect to the norm maps.We shall write Col u for the power series in A p ((x)) ∗ associated to u ∈ lim←K ∗ n by Coleman’sisomorphism of Theorem 6.2.Remark 6.3. An easily obtained family of N -invariant power series is the following. Leta ∈ A ∗ p: then∏Φ a (x + u) =∏(Φ a (x) + Φ a (u)) = Φ π (Φ a (x))u∈Φ[p]u∈Φ[p](since Φ a permutes elements in Φ[p]) and from Φ π Φ a = Φ a Φ π in A p {{τ}} it follows that Φ a (x)is invariant under the Coleman norm operator N . (As observed in [6, page 797], this justamounts to replacing ω with a · ω as generator of the Tate module.)Following [11], we define an action of Γ on F p [[x]] 1 by (σ ∗ f)(x) := f(Φ χ(σ) (x)). ThenCol σu = (σ ∗ Col u ), as one sees from(6.2) (σ ∗ Col u )(ω n ) = Col u (Φ χ(σ) (ω n )) = Col u (σω n ) = σ(Col u (ω n )) = σ(u n ).6.3. The Coates-Wiles homomorphisms. We introduce some operators on power series.Let dlog : F p ((x)) ∗ 1 → F p((x)) 1 be the logarithmic derivative, i.e., dlog (g) := g′g. Also, forany j ∈ N let ∆ j : F p ((x)) → F p ((x)) be the jth Hasse-Teichmüller derivative, defined by theformula( ∞)∑∞∑( )∆ j c n x n n + j:=cj n+j x nn=0d jn=0(i.e., ∆ j “is” the differential operator 1 j!). A number of properties of the Hasse-Teichmüllerdx jderivatives can be found in [24]; here we just recall that the operators ∆ j are F p -linear andthat∞∑(6.3) f(x) = ∆ j (f) |x=0 x j .j=0The last operator we need to introduce is composition with the Carlitz exponential e C (x) =x + . . . , i.e., f ↦→ f(e C (x)).Definition 6.4. For any integer k ≥ 1, define the kth Coates-Wiles homomorphism δ k : lim←O ∗ n →F p byδ k (u) := ∆ k−1((dlog Colu )(e C (x)) ) |x=0 = ( ∆ k−1 ((dlog Col u ) ◦ e C ) ) (0) .Notice that by (6.3) this is equivalent to putting∞∑(6.4) (dlog Col u )(e C (x)) = δ k (u)x k−1 .k=1Lemma 6.5. The Coates-Wiles homomorphisms satisfyδ k (σu) = χ(σ) k δ k (u).Proof. Recall that ddx Φ a(x) = a for any a ∈ A p . Then from (6.2) it followsdlog Col σu = dlog (Col u ◦ Φ χ(σ) ) = χ(σ)(dlog Col u ) ◦ Φ χ(σ) ,
since dlog (f ◦ g) = g ′ ( f ′fby (6.4),IWASAWA THEORY OVER FUNCTION FIELDS 25◦ g). Composing with e C and using Φ a (e C (x)) = e C (ax), one gets,(dlog Col σu )(e C (x)) = χ(σ)(dlog Col u )(e C (χ(σ)x)) = χ(σ)The result follows.6.4. Cyclotomic units.∞∑δ k (u)χ(σ) k−1 x k−1 .Definition 6.6. The group C n of cyclotomic units in F n is the intersection of B ∗ n with thesubgroup of F ∗ n generated by σ(ω n ), σ ∈ Gal(F n /F ).By the explicit description of the Galois action via Φ, one sees immediately that this is thesame as B ∗ n ∩ 〈Φ a (ω n )〉 a∈A−p .Lemma 6.7. Let ∑ c σ σ be an element in Z[Gal(F n /F )]: then∏σ(ω n ) cσ ∈ C n ⇐⇒ ∑ c σ = 0 .σ∈Gal(F n /F )Proof. Obvious from the observation that ω n is a uniformizer for the place above p and a unitat every other finite place of F n .□Let C n and C 1 n denote the closure respectively of C n ∩ O ∗ n and of C 1 n := C n ∩ U n .Let a ∈ A ∗ p. By Remark 6.3 (Φ a (ω n )) n is a norm compatible system: hence one can definea homomorphismΥ: Z[˜Γ] −→ lim K ∗←n∑cσ σ ↦→ ∏ (σ(ωn ) c ) σn = ∏ (Φχ(σ) (ω n ) c ) σnLet lim ̂ K ∗←n be the p-adic completion of lim K ∗←n. By (6.1) one gets the isomorphism lim ̂ K ∗←n ≃ω Z p× lim←U n .Lemma 6.8. The restriction of Υ to Z[Γ] can be extended to Υ: Λ → ̂ lim←K ∗ nProof. If a ∈ A p is a 1-unit, then(6.5) Φ a (ω n ) = ω n u nwith u n ∈ U n . Since by definition Γ = χ −1 (1 + πA p ), it follows that Υ sends Z[Γ] intoω Z × lim←U n . To complete the proof it suffices to check that Υ is continuous with respect tothe natural topologies on Λ = lim(Z/p n Z)[Gal(F n /F )] and lim ̂ K ∗← ←n. But a ≡ a ′ mod π n in A pimplies Φ a (ω j ) = Φ a ′(ω j ) for any j ≤ n; and the result follows from the continuity of χ. □Proposition 6.9. Let I ⊂ Λ denote the augmentation ideal: then Υ induces a surjectivehomomorphism of Λ-modules I −→ lim←C 1 n. The kernel has empty interior.Proof. From Lemma 6.7 and (6.5) it is clear that Υ(α) ∈ lim←C 1 n if and only if α ∈ I. Thismap is surjective because I is compact and already the restriction to the augmentation idealof Z[Γ] is onto Cn 1 for all n. A straightforward computation shows that it is a homomorphismof Λ-algebras: for γ ∈ Γ(∑ ) ( (∏ ) ) (∏ )(6.6) γΥ cσ σ = γ Φχ(σ) (ω n ) c σ= Φχ(σγ) (ω n ) c σnnbecause γ(Φ a (ω n )) = Φ a (γ(ω n )) = Φ a (Φ χ(γ) (ω n )).k=1□
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Page 11 and 12: Note that π F d= π F 1◦ π F dF
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