ASPECTS OF IWASAWA THEORY OVER FUNCTION FIELDS 1 ...

IWASAWA THEORY OVER FUNCTION FIELDS 23We putF n := F (Φ[p n ])andK n := F p (Φ[p n ]) .As stated in §1.2, we think of the F n ’s as subfields of C p , so that the K n ’s are their topologicalclosures. We shall denote the ring of A-integers in F n by B n and its closure in K n by O n , andwrite U n for the 1-units in O n . Let F := ∪F n and ˜Γ := Gal(F/F ).Consider the ring of formal skew power series A p {{τ}}: it is a complete local ring, withmaximal ideal πA p +A p {{τ}}τ. It is easy to see that Φ extends to a continuous homomorphismΦ: A p → A p {{τ}} (i.e., a formal Drinfeld module) and this allows to define a “cyclotomic”character χ: ˜Γ −→ ∼ A ∗ p. More precisely, let T p Φ := lim Φ[p n ] (the limit is taken with respect to←x ↦→ Φ π (x)) be the Tate module of Φ. The ring A p acts on T p Φ via Φ, i.e., a·(u) n := (Φ a (u n )) n ,and the character χ is defined by σu =: χ(σ) · u, i.e., χ(σ) is the unique element in A ∗ p suchthat Φ χ(σ) (u n ) = σu n for all n. From this it follows immediately that ˜Γ = ∆ × Γ, where∆ ≃ F ∗ p is a finite group of order prime to p and Γ is the inverse image of the 1-units.Since Φ has rank 1, T p Φ is a free A p -module of rank 1. As in [6], we fix a generatorω = (ω n ) n≥1 : this means that the sequence {ω n } satisfiesΦ π n(ω n ) = 0 ≠ Φ π n−1(ω n ) and Φ π (ω n+1 ) = ω n .By definition K n = F p (ω n ). By Hayes’s theory, the minimal polynomial of ω n over Fis Eisenstein: it follows that the extensions F n /F and K n /F p are totally ramified, ω n is auniformizer for the field K n , O n = A p [[ω n ]] = A p [ω n ]. The extension F n /F is unramified atall other finite places: this can be seen directly by observing that Φ π n has constant coefficientπ n . Furthermore F n /F is tamely ramified at ∞ with inertia group I ∞ (F n /F ) ≃ F ∗ q.The similarity with the classical properties of Q(ζ p n)/Q is striking.The formula N Fn+1 /F n(ω n+1 ) = ω n shows that the ω n ’s form a compatible system underthe norm maps (the proof is extremely easy; it can be found in [6, Lemma 2]). This and theobservation that [F n+1 : F n ] = q deg(p) for n ≥ 1 imply(6.1) lim←K ∗ n = ω Z × F ∗ p × lim←U n .Note that lim←U n is a ˜Λ-module.6.2. Coleman’s theory. A more complete discussion and proofs of results in this sectioncan be found in [6, §3]. Let R be a subring of C p : then, as usual, R((x)) := R[[x]](x −1 ) is thering of formal Laurent series with coefficients in R. Moreover, following [11] we define R[[x]] 1and R((x)) 1 as the subrings consisting of those (Laurent) power series which converge on thepunctured open ballB ′ := B(0, 1) − {0} ⊂ C p .The rings R[[x]] 1 and R((x)) 1 are endowed with a structure of topological R-algebras, inducedby the family of seminorms {‖ · ‖ r }, where r varies in |C p | ∩ (0, 1) and ‖f‖ r := sup{|f(z)| :|z| = r}.All essential ideas for the following two theorems are due to Coleman [11].Theorem 6.1. There exists a unique continuous homomorphismsuch thatN : F p ((x)) ∗ 1 → F p ((x)) ∗ 1∏f(x + u) = (N f) ◦ Φ π .u∈Φ[p]