arXiv:math/0702441v2 [math.NT] 6 Feb 2009 COLEMAN’S POWER SERIES AND WILES’ RECIPROCITY FOR RANK 1 DRINFELD MODULES FRANCESC BARS, IGNAZIO LONGHI Abstract. We introduce the formalism of Coleman’s power series for rank 1 Drinfeld modules and apply it to formulate and prove the analogue of Wiles’ explicit reciprocity law in this setting. 1. Introduction Andrew Wiles discovered an explicit reciprocity law for local fields [16], generalizing earlier work of Artin-Hasse [2] and Iwasawa [8]. Since the publication of [16], different proofs of this reciprocity law were found: see the expositions in [10, chapter 9], [5, I, §4] (where the main tool are Coleman’s power series) and [13, §3.3] (a cohomological approach, inspired by Kato’s formulation in [9, §1]). Wiles’ explicit reciprocity law has its foundation in the theory of Lubin-Tate formal groups: torsion points of the formal group generate a “cyclotomic” tower of local fields and the action of local norm symbols on the torsion is expressed by an analytic formula. A rank 1 Drinfeld module of generic characteristic can be seen as originating a special instance of Lubin-Tate formal group: therefore there should be an analogue of Wiles’ reciprocity law in this setting. For the Carlitz module, this was proven by Anglès: his paper [1] follows the approach of [10, chapter 9]. In the present paper we introduce the formalism of Coleman’s power series in the additive setting of (formal) rank 1 Drinfeld modules. As a first application we obtain our main result, the explicit reciprocity law for Drinfeld modules over any global field F (Theorem 24): i.e., we work with any sign-normalized rank 1 Drinfeld module, with no restriction on the class number. Following [5, I, §4] we formulate the law directly in its limit form, as the equality of two pairings on systems of groups. In the classical situation over Qp , one can exploit the Coleman isomorphism (corresponding to Theorem 11 in our setting) to construct p-adic L-functions: e.g., the Kubota-Leopoldt zeta function comes from the system of cyclotomic units. We hope to be able to recover this aspect of the theory in a future paper. Let us finally give a rough sketch of the contents of this paper. In §2 we introduce the basic notation and properties for formal Drinfeld modules and the “cyclotomic” tower we are working with. In §3 we rewrite Coleman’s formalism in the setting of Drinfeld modules: our construction is quite detailed, in the hope it can be a good Date: February 6, 2009. Both authors have been supported by TMR Arithmetic Algebraic Geometry. The first author is also supported by MTM2006-11391; the second one by a scholarship of Università di Milano. 1 2 FRANCESC BARS, IGNAZIO LONGHI introduction to the subject (the differences with characteristic 0 are irrelevant). In §4 we introduce the two pairings and prove their equality. 2. Setting Let F be a global function field, with field of constants Fq , q a power of p. Given a place v of F , qv := q deg(v) will indicate the cardinality of its residue field Fv . Besides, we denote by τ the operator x 7→ xq and, for an Fp -algebra R, by R{τ } the ring of skew polynomials with coefficients in R: multiplication in R{τ } is given by composition. 2.1. Review of rank 1 Drinfeld modules. We briefly recall Hayes’ theory of explicit construction of class fields by means of rank 1 Drinfeld modules; our main references will be [7] and [6, chapter 7]. We fix a place ∞ of F and let A ⊂ F be the ring of functions regular away from ∞. As in [7] and [6] we fix a sign-function sgn : F∞ → F∞ : then the basic extensions of F are H and H + , the Hilbert class field and the normalizing field (see e.g. [7, §14-15] or [6, §7.1 and §7.4]), with B and B + the integral closure of A in H and H + respectively. (It might be worth to recall that H depends only on the choice of ∞, while H + is determined by both ∞ and sgn.) Let Φ be a sgn-normalized rank 1 Drinfeld A-module: i.e., Φ is a ring homomorphism A → B + {τ }, a 7→ Φa , such that the constant coefficient of Φa is a and the leading coefficient map is a Gal(F∞ /Fq )-twist of sgn. We fix such a Φ. As usual, if I is an ideal of A, Φ[I] denotes the I-torsion of Φ (i.e., the common zeroes of all Φa , a ∈ I) and ΦI is the unique monic generator of the left ideal of H + {τ } generated by Φa , a ∈ I. One sees immediately that Y (x − u) ΦI (x) = u∈Φ[I] and since elements in Φ[I] are all integral above B + , it follows that ΦI ∈ B + {τ }. By [7, Proposition 11.4], in case I = pn , p a prime, all irreducible factors (over B + ) of the polynomial ΦI (x) are Eisenstein at p. To conclude, we recall that the extension H + (Φ[pn ])/F is abelian and Gal(H + (Φ[pn ])/H + ) ≃ (A/pn )∗ ([6, §7.5]): the isomorphism is given by the A-action on Φ[pn ]. Caveat. The notation Φa (or ΦI ) will be used to denote both the operator Φa ∈ B + {τ } and the polynomial Φa (x) ∈ B + [x]; the context should make clear which one we mean. 2.2. Our local setting. From now on, we fix a prime ideal p in A: since Φ is sgn-normalized, it has good reduction in p. Let Fp and Ap be the completions at p. We also fix Cp , completion of an algebraic closure of Fp , and choose an embedding H + ֒→ Cp ; let K be the topological closure of H + in Cp and O = OK its ring of integers, with maximal ideal m = mK . Let M := B(0, 1) = {z ∈ Cp : |z| < 1}. In what follows, all extensions of F are assumed to be contained in Cp . The valuation v on Cp is normalized so that v(Fp∗ ) = Z. COLEMAN’S POWER SERIES 3 The extension H + /F is unramified outside of ∞: in particular, it follows that K/Fp is unramified. Let I be the group of fractional ideals of A and P + the subgroup of the positively generated principal ones. By Hayes’ theory we know that I/P + ≃ Gal(H + /F ) ([7, Theorem 14.7]); this isomorphism is given by the action of ideals on the set of sgn-normalized rank 1 Drinfeld modules. In particular the image of p in I/P + corresponds to the Frobenius at p: its order is f := [K : Fp ], hence we have pf = (η), with η ∈ A a positive element (i.e., sgn(η) = 1). Then, by definition of sgn-normalized, Φη is monic and therefore Φη = Φpf . Notice that η is uniquely defined once we have fixed sgn. Finally, we remark that, since all factors of Φpn are Eisenstein, Φ[pn ] ⊂ M for all n. In particular, all coefficients (but the leading one) of Φη are in m. 2.3. The formal module. Consider the ring of skew power series O{{τ }}: it is a local ring, complete in the topology induced by its maximal ideal m + O{{τ }}τ . As observed by Rosen ([11, p. 247]), the homomorphism Φ : A → B + {τ } ⊂ O{{τ }} can be extended to the localization of A at p and then to its completion: we get a formal Drinfeld module Φ : Ap → O{{τ }}. Proposition 1 (Rosen). There exists a unique λ ∈ K{{τ }} of the form λ = 1 + ... and such that aλ = λΦa for all a ∈ Ap . Besides, λ converges on M and it enjoys the following properties: P (1) λ = ci τ i with v(ci ) ≥ −i; (2) if v(x) > (q − 1)−1 , then v(λ(x)) = v(x). For the proof the reader is referred to [11, Proposition 2.1 and Proposition 2.3]. 2.4. The “cyclotomic” tower. Define the tower of field extensions Kn /K by Kn := K(Φ[pf n ]), n ≥ 1, with ring of integers On and maximal ideal mn . (This choice of indexing seemed to us more convenient even if slightly unusual - comn monly, one calls the tamely ramified extension K0 .) Let T rm : Kn → Km and ∗ ∗ n Nm : Kn → Km denote respectively trace and norm. We also put K∞ := ∪∞ n=1 Kn . Consider the Tate module Tp Φ := lim Φ[pf n ] (the limit is taken with respect to ← x 7→ Φη (x)).1 The ring Ap acts on Tp Φ via Φ: i.e., a · u := Φa (u). Since Φ has rank 1, Tp Φ is a free Ap -module of rank 1. Let ω = {ωn }n≥1 be a generator: Tp Φ = Ap · ω. This means that the sequence {ωn } satisfies Φnη (ωn ) = 0 6= Φηn−1 (ωn ) and Φη (ωn+1 ) = ωn . (Here and in the following, when we write Φna the power is always taken in O{τ } that is, with respect to composition.) By definition Kn = K(ωn ). Being a root of an Eisenstein polynomial, ωn is a uniformizer for the field Kn : it follows that the extensions Kn /K are totally ramified, Gal(Kn /K) ≃ (A/pf n )∗ and On = O[[ωn ]] = O[ωn ] . n 1More canonically, T Φ is usually defined as Hom p Ap (Fp /Ap , lim Φ[p ]) ([6, Definition 4.10.9] → and following remarks), but the two are isomorphic and our definition suits better our purpose. 4 FRANCESC BARS, IGNAZIO LONGHI The Galois action on Tp Φ is via the “Carlitz-Hayes” character χ : GK → A∗p , defined by σω = χ(σ) · ω for σ ∈ GK := Gal(Ksep /K): that is, χ(σ) is the unique element in A∗p such that Φχ(σ) (ωn ) = σωn for all n. Lemma 2. The elements ωn form a compatible system under the norm maps: Nnn+1 (ωn+1 ) = ωn . Proof. Either [Kn+1 : Kn ] is odd or the characteristic is 2: in both cases, it follows that −Nnn+1 (ωn+1 ) is the constant term of the minimal polynomial of ωn+1 on Kn . It is immediate to see that the latter is Φη (x) − ωn (it is Eisenstein).  We also remark that the Gal(Kn+i /Kn )-orbit of ωn+i is exactly ωn+i + Φ[pif ]. Lemma 3. Let Dn := DKn /Fp be the different of Kn /Fp : then Dn is generated by an element of valuation 1 . nf − qp − 1 Moreover, ωn has valuation v(ωn ) = [Kn : K]−1 = qp1−nf (qp − 1)−1 . Proof. The last assertion is obvious from the already remarked fact that ωn is a uniformizer in a totally ramified extension. As for the first, let ψn be the irreducible polynomial of ωn over H + , n ≥ 1: then Dn = DKn /K because K/Fp is unramified and since DKn /K = (ψn′ (ωn )) we just need to compute this derivative. From [7, Proposition 11.4] we get the equality in B + [x] ψn (x)Φpf n−1 (x) = Φpf n (x) = Φηn (x) . Differentiating and evaluating in ωn we get v(ψn′ (ωn )) = nf − v(Φpf n−1 (ωn )) = nf − (qp − 1)−1 . (Observe that Φpf n−1 (ωn ) has valuation (qp − 1)−1 because Φpf n−1 (x) is a monic polynomial of degree qpnf −1 all whose coefficients but the leading one are in mK .)  Corollary 4. Let T rn : Kn → Fp be the trace map. Then v(T rn (x)) ≥ ⌊v(x) + nf − (qp − 1)−1 ⌋ where ⌊r⌋ denotes the largest integer ≤ r. Similarly, for m ≥ 1, n v (T rm (x)) > v(x) + (n − m)f − v(ωm ) . Proof. Let k = ⌊v(x) + nf − (qp − 1)−1 ⌋ : then xOn ⊆ pk D−1 n by Lemma 3 and this means that T rn (xOn ) ⊆ pk Ap , by a basic property of the different (see e.g. [14, III, §3, Proposition 7]). In the same way, using the fact (obvious from Lemma 3) that the generator of DKn /Km has valuation (n − m)f , one gets n v (T rm (x)) ≥ the second statement follows.  v(x) + (n − m)f  v(ωm ) ; v(ωm )  COLEMAN’S POWER SERIES 5 2.5. Local class field theory. The Carlitz-Hayes character induces an isomorphism of topological groups χ−1 : A∗p → Gal(K∞ /K) (recall that we put K∞ = ∪∞ n=1 Kn ). This should be compared with the local norm symbol map ( · , K∞ /Fp ) : Fp∗ → Gal(K∞ /Fp ) . By class field theory, the image of A∗p in Gal(K∞ /Fp ) is exactly Gal(K∞ /K). Let a ∈ A be a generator of a prime ideal q 6= p and assume sgn(a) = 1, so that Φa = Φq ; moreover, let F robq ∈ Gal(H + (Φ[p∞ ])/F ) denote the Frobenius at q. Passing from local to global class field theory one finds F robq = (a−1 , K∞ /Fp ). By [6, Proposition 7.5.4] Φa acts on ω as F robq and hence χ−1 (a) = (a−1 , K∞ /Fp ). By Tchebotarev for any n all elements in Gal(H + (Φ[pn ])/H + ) can be represented as F robq for some q as above. Therefore the corresponding a’s are dense in A∗p and we get (1) (u, K∞ /K)(ω) = Φu−1 (ω) for all u ∈ A∗p . 3. Coleman’s formalism Let K be a local field and {Kn } the tower of extensions of K generated by the torsion of a Lubin-Tate formal module: in [4], Coleman discovered an isomorphism between lim Kn∗ and the subgroup of OK ((x))∗ fixed under a certain operator N . ← The same formalism can be used in the context of Drinfeld modules, as follows. 3.1. A bit of functional analysis. Let R be a subring of Cp : then, as usual, R((x)) := R[[x]](x−1 ) is the ring of formal Laurent series with coefficients in R. Moreover, following [4] we define R[[x]]1 and R((x))1 as the subrings consisting of those (Laurent) power series which converge on the punctured open ball B ′ := B(0, 1) − {0} ⊂ Cp . These latter rings are endowed with a structure of topological R-algebras, induced by the family of seminorms {k · kr }, where r varies in |Cp | ∩ (0, 1) and kf kr := sup{|f (z)| : |z| = r}. One easily checks that this is the same as the “compact-open” topology of [4, pag. 93]: in particular a sequence {fn } in O((x))1 converges to f ∈ O((x))1 if and only if for each closed annulus C around zero in B ′ and for each ǫ > 0 there exists a positive integer N (C, ǫ) such that |fn (a)−f (a)| < ǫ for all a ∈ C and all n ≥ N (C, ǫ). Let φ ∈ O{τ } be an additive polynomial having all its zeroes in M. One checks easily that, since |φ(z)| ≤ |z| for all z ∈ B ′ , the map ◦φ : g 7→ g ◦ φ defines a continuous endomorphism of K[[x]]1 . Lemma 5. Let φ be as above; furthermore assume that φ(x) is separable. Then the image of ◦φ : K[[x]]1 → K[[x]]1 consists exactly of those g such that g(x+ u) = g(x) for all u zeroes of φ. This is essentially Lemma 3 of [4]. 6 FRANCESC BARS, IGNAZIO LONGHI Proof. For u ∈ M let Tu be the automorphism of Cp [[x]]1 given by g 7→ g(x + u). The inclusion \ ker(Tu − id) ◦φ(K[[x]]1 ) ⊂ φ(u)=0 is clear. Vice versa, assume that f ∈ K[[x]]1 is Tu -invariant for all u’s. Let K[[x]]r consist of those power series converging on the closed ball B(0, r): then K[[x]]1 is the inverse limit of K[[x]]r for r < 1. The Weierstrass division Theorem holds in each K[[x]]r and reasoning as in [4, Lemma 3] one can find fi ’s such that f (x) = n−1 X fi (0)φ(x)i + fn (x)φ(x)n i=0 for all n ≥ 0. It remains to show that fn φn tends to zero: one proves inductively that kfn kr ≤ kf kr kφk−n r , which implies, for s > r,  n kφkr kfn φn kr ≤ kfn ks kφn kr ≤ kf ks kφks because kgkr ≤ kgks . To conclude notice that kφkr < kφks .  Remark. For the goals of this paper, it would have been enough to prove the weaker statement that (◦φ)(K[[x]]1 ) is the closure of the subspace of Tu -invariant polynomials. This can be done without Weierstrass theory, as follows. Suppose f ∈ K[x] is Tu -invariant for all the zeroes of φ: we can assume inductively (taking as first step the constants) that if g ∈ K[x] enjoys this property and deg(g) < deg(f ) then g belongs to the image of ◦φ. By the euclidean algorithm for K[x], f = f1 φ + r; evaluation in the zeroes of φ shows that r = f (0) is a constant and then it is immediate to check that f1 is Tu -invariant. Finally observe that K[x] is dense in K[[x]]1 and the maps are all continuous. Corollary 6. The map ◦φ induces an isomorphism of topological algebras between K[[x]]1 and its image. Proof. Observe that K[[x]]1 is a Fréchet space over K (for definitions and basic properties, see e.g. [12, I,§8]). Lemma 5 implies that ◦φ(K[[x]]1 ), being closed, is Fréchet as well. Since ◦φ is injective, the corollary follows from the open mapping Theorem, as in [12, Corollary 8.7].  3.1.1. Some topological rings. To enhance clarity, we add a brief digression on topological structures for rings of power series. As above R is a subring of Cp . We let R[[x]] ≃ RN be a topological R-algebra with the product topology: that is, a fundamental system of neighbourhoods of 0 is given by nX o Uε,n := ai xi ∈ R[[x]] : |ai | < ε ∀ i < n . When we just write R((x)), we think of it as the additive group with the topology induced by the one on R[[x]] (observe however that with this topology R((x)) is not exactly a topological ring, since multiplication by x−1 is not continuous). To compare structures remember that if {fn } converges to f in R[[x]]1 then the individual coefficients of the power series fn converge to those of f : it follows that the inclusion R[[x]]1 ֒→ R[[x]] is continuous. Observe, however, that R((x))1 is not continuously injected in R((x)) : e.g., if |a| < 1 then an! x−n converges to 0 COLEMAN’S POWER SERIES 7 in O((x))1 , but not in O((x)) . This example shows as well that R((x))1 is not complete. When R is a subring of the closed ball B(0, 1) ⊂ Cp , R[[x]]1 ≃ R[[x]] as topological spaces and R((x))1 = R((x)) as sets. Moreover in this case we can furnish R((x)) with a third P topology, defined (if the restriction of v to R is discrete) by the valuation ν( ai xi ) := mini {v(ai )}. Once again, continuity of the inclusion (R[[x]], ν) ֒→ R[[x]]1 fails to extend to Laurent series. Lastly we remark that, for φ as above, ◦φ is a continuous endomorphism of K((x)) (but, of course, not of K((x))1 , unless φ has no zeroes in B ′ ) and even an automorphism if φ(x) is separable. In fact φ(x) separable means that its degree 1 coefficient is not zero, hence one can find ψ ∈ K[[x]] such that ψ(φ(x)) = x. 3.2. Coleman’s Theorems. Theorem 7 (Coleman). There exist unique continuous operators T , N : K((x))1 → K((x))1 such that respectively X g(x + u) = (T g) ◦ Φη Y g(x + u) = (N g) ◦ Φη . u∈Φ[pf ] u∈Φ[pf ] Moreover, T is a homomorphism of the additive group K((x))1 and N of K((x))∗1 . Proof. On K[[x]]1 the theorem is an immediate consequence of Corollary 6: T P Q (respectively N ) is just the composition of (◦Φη )−1 with Tu (resp. Tu ). In order to extend T and N to K((x))1 , remember that Φη belongs to xO[x]: then, if g ∈ K((x))1 , for some i ≥ 0 one has Φη (x)i g ∈ K[[x]]1 and we put if T (g) := x−i T (Φη (x)i g), N (g) := x−qp N (Φη (x)i g). These are well-defined: e.g., if g ∈ K[[x]]1 , Y Y f qf Y Tu (g) = (xqp N g)◦Φη . N (Φη (x)g)◦Φη = Tu (Φη (x)g) = Tu (Φη )Tu (g) = Φηp Additivity of T and multiplicativity of N are immediate.  As usual, we call T and N respectively the Coleman trace and norm. Y Lemma 8. The equality N k g ◦ Φkη = g(x + u) holds for any g ∈ K((x))∗1 . u∈Φ[pf k ] Proof. Assume by induction that the statement is true up to k − 1. Let W ⊂ Φ[pf k ] be a set such that Φη : W → Φ[pf k−f ] is a bijection. We have the following equalities: N k g(Φkη (x)) = N k−1 N g(Φηk−1 (Φη (x))) = Y Y = N g(Φη (x) + v) = N g(Φη (x + w)) = w∈W v∈Φ[pf k−f ] = Y w∈W (N g ◦ Φη )(x + w) = Y Y W u∈Φ[pf ] g(x + u + w) = Y g(x + t) . t∈Φ[pf k ]  8 FRANCESC BARS, IGNAZIO LONGHI Lemma 9. One computes: (N k g)(ωn ) = Nnn+k (g(ωn+k )). Similarly, T k g(ωn ) = T rnn+k (g(ωn+k )). Proof. Replace ωn = Φkη (ωn+k ) and apply Lemma 8.  Lemma 10. The restriction to O((x))∗1 of the sequence of operators N k converges to a continuous endomorphism N ∞ . Proof. Observe that O((x))∗1 = xZ × O[[x]]∗ and that N x = x. Therefore the lemma is proven if we show that, for any g ∈ O[[x]]∗ , N k g is a Cauchy sequence with respect to the valuation topology on O[[x]] (uniformly on g). More precisely, we are going to prove that N k+1 g ≡ N k g mod mk+1 by induction on k. K First notice that Y f Φη (x) = (x − u) ≡ xqp mod mK u∈Φ[pf ] since v(u) > 0 for u ∈ Φ[pf ]. Therefore Y f f f g qp ≡ g(x + u) = N g ◦ Φη ≡ N g(xqp ) ≡ (N g(x))qp mod mK where the last congruence is true because qpf = |OK /mK |. It follows that N g ≡ g mod mK . k Now put h := NNk−1g g , so that our claim becomes N h ≡ 1 mod mk+1 K . By the induction hypothesis h = 1 + π k g1 and this implies Y f (N h ◦ Φη )(x) = (1 + π k g1 (x + u)) ≡ (1 + π k g1 (x))qp ≡ 1 mod mk+1 K . u∈Φ[pf ] To conclude, observe that since Φη is monic ν(h ◦ Φη ) = ν(h) for any h (where ν is the valuation on O[[x]] defined in §3.1.1).  Of course N ◦ N ∞ = N ∞ and N ∞ is a projection. Theorem 11 (Coleman). The evaluation map ev : f 7→ {f (ωn )} gives an isomorphism (O((x))∗ )N =id ≃ lim Kn∗ ← where the inverse limit is taken with respect to the norm maps. Proof. The map is injective, because a function is uniquely determined by its values at the ωn ’s (e.g., observe that |ωn | < 1 and use [6, Proposition 2.11]). Notice that O((x))∗ = xZ × O[[x]]∗ and lim Kn∗ = ω Z × lim On∗ : since ev(x) = ω, ← ← it suffices to show (O[[x]]∗ )N =id ≃ lim On∗ . ← Consider the diagram Q ∗ ev O[[x]]∗ −−−−→ On   N /id N/id y y Q ev On∗ O[[x]]∗ −−−−→ where N is the norm map (xn ) 7→ (Nnn+1 xn+1 ). It commutes by Lemma 9; lim On∗ ← is the kernel of the right-hand side: hence ev(g) ∈ lim On∗ iff N g = g. Since ← COLEMAN’S POWER SERIES 9 (O[[x]]∗ )N =id = N ∞ (O[[x]]∗ ) is compact (because so is O[[x]]∗ ), the theorem is proven if we show that the image of ev is dense in a set containing lim On∗ . For any ← u = (un ) ∈ lim On∗ and any k there exists g ∈ O[[x]]∗ such that g(ω2k ) = u2k . Let ← h := N k g. Remembering (from the proof of Lemma 10) that N k g ≡ N k+r g mod mk for any r ≥ 0, we get h(ωi ) ≡ N 2k−i g(ωi ) = Ni2k (g(ω2k )) = ui mod mk for all i = 1, ..., k: density follows.  In particular, ω ∈ lim On∗ corresponds to x ∈ O((x))∗ and, more generally, for ← a ∈ A∗p the element a · ω corresponds to Φa (x) ∈ O((x))∗ (this is equivalent to changing generator of Tp Φ). We conclude with a lemma we are going to use in the next section. Lemma 12. Let dlog : K((x))∗1 → K((x))1 be the logarithmic derivative operator, ′ dlog (g) := gg . Then T dlog g = ηdlog N g. Proof. One just computes: X Y (T dlog g)◦Φη = Tu (dlog g) = dlog Tu (g) = dlog (N g◦Φη ) = η(dlog (N g)◦Φη ) using the fact that dlog is a homomorphism and d dx Φη (x) = η.  3.3. Higher rank Drinfeld modules. As the reader may have noticed, the rank of the Drinfeld module Φ plays essentially no part in Theorem 7. This suggests the possibility of extending Coleman’s results to Drinfeld modules of any rank: we sketch an approach. In this subsection (and only here) our notations are slightly modified. For simplicity we take F := Fq (T ) and A := Fq [T ] : then, fixing a prime ideal p = (π) of A, we have K = Fp and O = Ap . Let Φ : A → O{τ } be a rank r Drinfeld module, r > 1; we need the hypothesis that Φ has good reduction mod mK and that the height is maximal: h = r. In particular, it follows that all zeroes of Φπ are in the open unit ball B(0, 1) ⊂ Cp . We also assume that Φπ (x) is a monic polynomial. Then, reasoning exactly as in Theorem 7 and Lemma 10, one proves the following. Theorem 13. There exists a continuous homomorphism N : K((x))∗1 → K((x))∗1 such that Y g(x + u) = (N g) ◦ Φπ . u∈Φ[p] to O((x))∗1 ∞ The restriction endomorphism N . of the sequence of operators N k converges to a continuous As in section 2.4, we choose a sequence {ωn }n≥1 so that Φnπ (ωn ) = 0 6= Φπn−1 (ωn ) and Φπ (ωn+1 ) = ωn and construct a tower {Kn } by Kn := Kn−1 (ωn ), with K0 := K. Because of the rank, these extensions are much smaller than K(Φ[pn ]) and they are not Galois; however, since the polynomials Φπ (x)x−1 and Φπ (x) − ωn are Eisenstein, it still 10 FRANCESC BARS, IGNAZIO LONGHI holds that each Kn /K is totally ramified, with uniformizer ωn . For any n ≥ 1 there is a norm map Y ∗ Nnn+1 : Kn+1 → Kn∗ , a 7→ σ(a) σ∈Sn+1 where the product is taken on the set of embeddings Sn+1 := {σ : Kn+1 ֒→ Cp : σ|Kn = idKn } . It follows from the additivity of Φπ that the assignment σ 7→ σ(ωn+1 ) − ωn+1 is a bijection Sn+1 → Φ[p] : therefore Nnn+1 (ωn+1 ) = ωn and, more generally, Y g(ωn+1 + u) = Nnn+1 (g(ωn+1 )) . (N g)(ωn ) = N g(Φπ (ωn+1 )) = u∈Φ[p] Theorem 14. The evaluation map ev : f 7→ {f (ωn )} gives an isomorphism (O((x))∗ )N =id ≃ lim Kn∗ . ← The proof is the same as for Theorem 11. 4. The explicit reciprocity law The reader is reminded that T rn , Nn denote respectively trace and norm from Kn to Fp . Also, we let (·, Lab /L) : L∗ −→ Gab L be the local norm symbol map and write Colu for the power series in O((x))∗ associated to u ∈ lim Kn∗ by Coleman’s isomorphism of Theorem 11. To lighten ← notation, in this section the action of Ap via Φ will be often denoted by a·x := Φa (x). 4.1. The Kummer pairing. Next to lim Kn∗ we consider lim Kn , defined as the ← → direct limit of the maps Φη : Kn → Kn+1 : that is, lim Kn consists of sequences → a = (an )n≥N (for some N ∈ N) such that aN ∈ KN and an+1 = Φη (an ), modulo the relation (an )n≥N = (bn )n≥M if an = bn for n ≫ 0. Kummer theory yields a pairing ( , )n : Kn × Kn∗ → Φ[pf n ] defined by √ (a, u)n := ((u, Knab /Kn ) − 1)( ηn a). √ Here ηn a is a solution of Φnη (x) = a: since any two roots differ by an element in √ Φ[pf n ] ⊂ Kn , the value of (a, u)n is independent of the choice of ηn a. p √ Observe that, since by definition ηn a =n+1 η Φη (a), q ab (Φη (a), u)n+1 = ((u, Kn+1 /Kn+1 ) − 1)(n+1 η Φη (a)) = √ = ((Nnn+1 (u), Knab /Kn ) − 1)( ηn a) = (a, Nnn+1 (u))n . This means that, given a = (an ) ∈ lim Kn and u = (un ) ∈ lim Kn∗ , one has (for any → ← n large enough that an exists) (an+1 , un+1 )n+1 = (an , un )n . Therefore we can define a limit form of the Kummer pairing ( , ) : lim Kn × lim Kn∗ → Φ[p∞ ] → by (a, u) := (an , un )n for n ≫ 0. ← COLEMAN’S POWER SERIES 11 One checks immediately that ( , ) is bilinear, additive in the first variable and multiplicative in the second. In particular, since values are in a group of exponent p, it follows that (·, ζ)n = 0 for any root of unity ζ ∈ Kn∗ . Lemma 15. All the pairings ( , )n are continuous. Furthermore, (a, ·)n ≡ 0 for any a ∈ Kn such that v(a) > nf + (qp − 1)−1 . Proof. For any a ∈ Kn , (a, ·)n is continuous: therefore the first assertion follows from the second, which in√turn is an easy application of Krasner’s Lemma. In fact, if one can choose α = ηn a so that v(α) is big enough, then it follows (a, u) = 0 because |(a, u)| ≤ |α| = |(u, Knab /Kn )(α)| and Φ[pnf ] is a discrete subset of C∞ . We are left with a valuation computation. First of all, observe that for any j ≥ 1 if u ∈ Φ[pj ] − Φ[pj−1 ] then vj := v(u) = [K(Φ[pj ]) : K]−1 = |(A/pj )∗ |−1 = (because Φpj (x) Φpj−1 (x) 1 qpj−1 (qp − 1) is Eisenstein). In particular the smallest non-zero elements in Φ[pnf ] have valuation v1 = (qp − 1)−1 . We also put v0 := ∞. Now choose α a root of Φnη (x) = a such that v(α) is maximal: we get Y X a= (α + u) and v(a) = v(α + u) , u∈Φ[pf n ] u∈Φ[pf n ] with v(α + u) = min{v(α), v(u)} for all u’s because of the maximality hypothesis. Hence if vj ≥ v(α) > vj+1 we obtain X X v(a) = v(α) + v(u) = qpj v(α) + nf − j u∈Φ[pj ] u∈Φ[pnf ]−Φ[pj ]  and 1 + nf − j + (qp − 1)−1 = qpj vj + nf − j ≥ v(a) > nf − j + (qp − 1)−1 . √ For a more detailed analysis of how v(a) determines the extension K1 ( ηn a)/K1 see [1, Proposition 2.1]. We remark that the computation in the proof of Lemma 15 yields immediately the following result. Lemma 16. For any a = (an )n≥N ∈ lim mn there exists a constant c(a) such that → v(an ) ≥ nf − c(a) for all n ≥ N . Lemma 17. Let a ∈ Kn∗ : then (a, a)n = 0. √ Proof. Let α be a representative of ηn a and put L := Kn (α). Kummer theory identifies Gal(L/Kn ) with a subgroup V of Φ[pnf ]: then one sees that Y Y Y a= (α + u + v) = NL/Kn (α + u) u∈Φ[pnf ]/V v∈V Φ[pnf ]/V and consequently (a, Knab /Kn ) acts trivially on L. bc , b−1 )n for all c ∈ Kn . Lemma 18. Let b ∈ mn − {0}: then (c, 1 − b)n = ( 1−b  12 FRANCESC BARS, IGNAZIO LONGHI Proof. If c = 0 both sides are 0. If not, by applying Lemma 17 to a = c(1 − b) we get, by bilinearity, (c, 1 − b)n = (cb, c)n + (cb, 1 − b)n and, by recurrence, (c, 1 − b)n = ∞ X (cbj , cbj−1 )n j=1 (the sum converges because only a finite number of terms are not 0, by Lemma 15). By Lemma 17 we have (cbj , cbj−1 )n = (cbj , cbj )n + (cbj , b−1 )n = (cbj , b−1 )n and therefore ∞ ∞ X X cb bj , b−1 )n = ( (cbj , b−1 )n = (c (c, 1 − b)n = , b−1 )n . 1 − b j=1 j=1  Proposition 19. Let a ∈ lim mn , u ∈ lim Kn∗ : then → ← (a, u) = (an ωn dlog Colu (ωn ), ωn )n for all n sufficiently large. Proof. The statement is clearly true for u = ω. As for u ∈ lim On∗ , we are going to ← prove that, more generally,
dlog w , ωn n (c, w)n = cωn dωn for all (c, w) ∈ msnn × On∗ for some sn . Here dlog : On∗ → ΩOn /O is the map x 7→ dx x . The module of differentials is free over On /Dn with generator dωn : by Lemma 3 if c ∈ mn with v(c) > qp2−1 − v(ωn ) and δ ∈ Dn the inequality v(cωn δ) ≥ v(c) + v(ωn ) + nf − 1 1 > nf + qp − 1 qp − 1 w holds and so Lemma 15 shows that (cωn dlog dωn , ωn )n is well-defined. Thanks to Lemma 16 this is enough for our purposes. Observe that it suffices to prove the claim for w = 1−ζωnk (ζ varying among roots of unity in Kn ), because one can choose a topological basis of 1 + mn consisting of elements of this form and the pairing is continuous and linear. Applying Lemma 18 with x = ζωnk we get (c, 1 − ζωnk )n = ( because dlog (1 − ζωnk ) cζωnk −k , ω ) = (cω , ωn )n n n n 1 − ζωnk dωn dlog (1 − ζωnk ) = −kζωnk−1 dωn . 1 − ζωnk dlog un . (Caveat! In dωn this last formula the symbol dlog appears with two different meanings: on the leftd hand side dlog Colu is the power series Col1u (x) dx Colu (x), evaluated in ωn , while ∗ on the right-hand side dlog : On → ΩOn /O is the map we defined above.)  To conclude just notice that, for u ∈ lim Kn∗ , dlog Colu (ωn ) = ← COLEMAN’S POWER SERIES 13 4.2. The analytic pairing. As above, let u ∈ lim Kn∗ . Lemmata 9 and 12 together ← with the N -invariance of Colu yield T rnn+k dlog Colu (ωn+k ) = η k dlog Colu (ωn ). Besides, for a = (an ) ∈ lim mn , λ(an+k ) = η k λ(an ) by definition of λ (Proposition → 1). It follows that T rn+k (η −n−k λ(an+k )dlog Colu (ωn+k )) = η k T rn (η −n λ(an )dlog Colu (ωn )). Lemma 20. T rn (η −n λ(an )dlog Colu (ωn )) ∈ Ap for n ≫ 0. Proof. By Lemma 16, v(an ) ≥ nf − c for some constant c (depending on a): Proposition 1 implies that the same is true for λ(an ). Since Colu ∈ xZ × O[[x]]∗ one has v(dlog Colu (ωn )) ≥ −v(ωn ) and the last value, in turn, is controlled by Lemma 3. Now apply Corollary 4.  Therefore one can define a second pairing [ , ] : lim mn × lim Kn∗ → Φ[p∞ ] → ← putting [a, u] := T rn (η −n λ(an )dlog Colu (ωn )) · ωn for n ≫ 0. (Recall that, by definition, a · ωn = Φa (ωn ) .) 4.2.1. It is convenient to define also a level n pairing [ , ]n : mtnn × Kn∗ → Φ[pnf ] for some tn ≥ 1. The logarithmic differential we used in the proof of Proposition 19 can be extended to a homomorphism dlog : Kn∗ = ωnZ × On∗ → m−1 n /Dn dωn by putting dlog ωni := iωn−1 . dωn Lemma 21. The pairing   dlog u −n · ωn [a, u]n := T rn η λ(a) dωn is well defined for v(a) ≥ 2(q − 1)−1 . Proof. We have to show that T rn (η −n λ(a)b) belongs to Ap for any b ∈ m−1 n and that
v T rn (η −n λ(a)δ) ≥ nf for δ ∈ Dn . Since the hypothesis implies v(λ(a)) = v(a), both assertions are easy consequences of Lemma 3 and Corollary 4.  It is clear that if (a, u) ∈ lim mn × lim Kn∗ , the equality [a, u] = [an , un ]n holds → ← for n ≫ 0. Proposition 22. For n large enough, [a, u] = [an ωn dlog Colu (ωn ), ωn ]n . 14 FRANCESC BARS, IGNAZIO LONGHI Proof. To lighten notation, put D = ωn dlog Colu (ωn ); observe that D ∈ On . One has to check that i.e. that T rn (η −n λ(an )Dωn−1 ) · ωn = T rn (η −n λ(an D)ωn−1 ) · ωn , v T rn  λ(an D) − λ(an )D ωn By Proposition 1, 
≥ 2nf . ∞ X

i i ci aqn (Dq − D) ≥ min{q i v(an ) − i} v λ(an D) − λ(an )D = v i=1 i≥1 which for n ≫ 0 is at least 2nf − c for some c independent of n, by Lemma 16. Now apply Corollary 4 to get rid of c.  4.3. The reciprocity law. In this paragraph we prove Wiles’ explicit reciprocity law for rank 1 Drinfeld modules (Theorem 24). Proposition 23. Let a ∈ mn , v(a) ≥ 2(q − 1)−1 : then [a, ωn ]n = (a, ωn )n . Proof. Our proof is divided in many steps, along the lines of [5, I, §4]. −1 1. To start with, we put am := Φm−n (a) for all m ≥ n and let bm := am ωm . η Thanks to Lemmata 16 and 3 there is a constant c (depending only on a) such that v(bm ) ≥ mf − c. As a consequence of Lemma 17 we get 0 = (am + ωm , (1 + bm )ωm )m = (am , ωm )m + (am , 1 + bm )m + (ωm , 1 + bm )m for m ≥ n. 2. Claim: if m ≫ 0, then (am , 1 + bm )m = 0. Proof. By definition of the Kummer pairing, for m > n we have (am , 1 + bm )m = (an , Nnm (1 + bm ))n . Since 1 + bm tends to 1, so does also Nnm (1 + bm ) : the claim is proven because (a, ·)n is continuous and Φ[pnf ] discrete. 3. Claim: (an , ωn )n = ω2m − ΦNm (1+bm )−1 (ω2m ) for m ≫ 0. Proof. From the above, we have ab (an , ωn )n = (am , ωm )m = −(ωm , 1 + bm )m = (1 − (1 + bm , Km /Km ))( √ We can take ηm ωm = ω2m . The extension K2m /Fp is abelian: hence η √ m ωm ). ab (1 + bm , Km /Km )|K2m = (Nm (1 + bm ), Fpab /Fp )|K2m . Now one applies formula (1). 4. Claim: Nm (1 + bm )−1 ≡ 1 − T rm (bm ) mod p2mf for m big enough. Proof. Take k ∈ N such that kf > c + 1 where c is the constant which appeared in step 1. We can assume m ≫ k. m Put β := T rm−k (bm ): then v(β) ≥ v(bm ) + kf − v(ωm−k ) by Corollary 4. We have   1 m = 1−β +δ, Nm−k 1 + bm COLEMAN’S POWER SERIES 15 where δ is an element of Km−k such that v(δ) ≥ 2v(bm ). Now apply Nm−k to the above equation to obtain Nm (1+bm )−1 = Nm−k (1−β+δ) = 1−T rm−k (β−δ)+θ ≡ 1−T rm (bm ) mod p2mf , because summands in θ have valuation at least 2v(β − δ) ≥ 2 min{(m + k)f − c − v(ωm−k ), 2mf − 2c} and v(T rm−k (δ)) ≥ ⌊(3m − k)f − 2c − (qp − 1)−1 ⌋. −1 5. From steps 3 and 4 we get (a, ωn )n = T rm (am ωm ) · ω2m . On the other hand [a, ωn ]n = [(am ), ω] = [am , ωm ]m . Since Colω = x, dlog Colω = x1 and by definition we obtain       λ(am ) λ(am ) 1 λ(am ) m [am , ωm ]m = T rm m ·ωm = m T rm ·(η ·ω2m ) = T rm ·ω2m . η ωm η ωm ωm The proof is completed by the same reasoning as in Proposition 22.  Combining Propositions 19, 22 and 23, we get our reciprocity law: Theorem 24. The two pairings ( , ) and [ , ] on lim mn ×lim Kn∗ → Φ[p∞ ], defined → ← respectively by [a, u] := T rn (η −n λ(an )dlog Colu (ωn )) · ωn and √ (a, u) := ((un , Knab /Kn ) − 1)( ηn an ) for n ≫ 0, are equal. 4.3.1. The Kummer pairing in practice. A weaker form of our explicit reciprocity law can be used to calculate the Kummer pairing also when un ∈ Kn∗ is not a coordinate in an inverse limit or, even if it is the case, one does not know how to explicitly find Colu . Given un ∈ Kn∗ and an ∈ mn we want to compute (an , un )n . We need to impose that there exists um for some convenient m ≥ n such that Nnm (um ) = un . If so we have: (an , un )n = (Φη (an ), un+1 )n+1 = . . . = (Φjη (an ), un+j )n+j for any integer 0 ≤ j ≤ m − n. Put an+j := Φjη (an ). In particular suppose we can take m big enough to have v(am ) > 2/(q − 1) (and hence v(am ) > qp2−1 − v(ωn )). (An estimate on the required size of m − n can be obtained from the computations proving Lemma 16.) Then dlog um , ωm )m dωm (see the proof of Proposition 19) and by Proposition 23 we have (am , um )m = (am ωm (am , um )m = (am ωm dlog um dlog um , ωm )m = [am ωm , ωm ]m . dωm dωm By Proposition 1 we have v(λ(z) − z) ≥ mini≥1 {q i v(z) − i} ; in particular this minimum is attained in i = 1 when v(z) > 2(q − 1)−1 . Hence a simple computation
shows that the further condition v(am ) ≥ q1 mf + 1 + (qp − 1)−1 + v(ωm ) implies   am dlog um dlog um · ωm = [am , um ]m . , ωm ]m = T rm [am ωm dωm η m dωm 16 FRANCESC BARS, IGNAZIO LONGHI The limit form of the Kummer pairing, as in Theorem 24, is useful rather for the purposes of Iwasawa theory (which is not yet well understood in our setting) than for a concrete calculation of the pairing at a level n. Historically, the interest in computing the Kummer pairing for local fields at a finite level was originated by the study of diophantine equations, in particular the Fermat one. For a survey on various explicit reciprocity laws in the local case we refer to [3]. Acknowledgements The first idea of this paper was born when the authors met in Universität Münster, during fall 2000 (the first author was supported by SFB 478, the second one by the TMR network Arithmetic Algebraic Geometry for mobility of young researchers). The project was later carried on with support and/or hospitality by Dipartimento di Matematica of Università di Padova, Departament de Matemàtiques in the Universitat Autònoma de Barcelona, Dipartimento di Matematica of Università degli Studi di Milano and also Dipartimento di Matematica of Università di Pavia: we would like to thank all of these institutions. Thanks also to Matteo Longo, for his comments on a first draft of this paper. Finally we thank the referee for his or her comments and suggestions. References [1] B. Anglès: On explicit reciprocity laws for the local Carlitz-Kummer symbols. J. Number Theory 78 (1999), no. 2, 228–252. [2] E. Artin, H. Hasse: Die beiden Ergänzungssätze zum Reziprozitätsgesetz der ln -ten Potenzreste im Körper der ln -ten Einheitswurzeln. Abh. Math. Sem. Univ. Hamburg 6 (1928), 146-162. [3] F. Bars, I. Longhi: Reciprocity laws à la Iwasawa-Wiles. To appear in Bibl. Rev. Mat. Iberoamericana. [4] R. Coleman: Division values in local fields. Invent. Math. 53 (1979) 91-116. [5] E. de Shalit: Iwasawa theory of elliptic curves with complex multiplication. Perspectives in Mathematics, 3. Academic Press, Inc., Boston, MA, 1987. [6] D. Goss: Basic structures of function field arithmetic. Springer-Verlag, New York, 1996. [7] D. Hayes: A brief introduction to Drinfeld modules. In: The arithmetic of function fields (Columbus, OH, 1991), 1-32. de Gruyter, Berlin, 1992. [8] K. Iwasawa: On explicit formulas for the norm residue symbol. J. Math. Soc. Japan 20 (1968), 151-164. [9] K. Kato: Generalized explicit reciprocity laws. Adv. Stud. Contemp. Math. (Pusan) 1 (1999) 57-126 [10] S. Lang: Cyclotomic Fields I and II. GTM 121, Springer-Verlag, New York/Berlin, 1990. [11] M. Rosen: Formal Drinfeld modules. J. Number Theory 103 (2003), no. 2, 234-256. [12] P. Schneider: Nonarchimedean functional analysis. Springer-Verlag, Berlin, 2002. [13] A.J. Scholl: An introduction to Kato’s Euler systems. In: Galois representations in arithmetic algebraic geometry (A.J. Scholl and R.L. Taylor eds.) London Math. Soc. Lecture Notes 254 (1998) 379-460 [14] J.-P. Serre: Local Fields. GTM 67, Springer-Verlag, New York/Berlin, 1979. [15] L. Washington: Introduction to cyclotomic fields. Second edition. GTM 83, Springer-Verlag, New York, 1997 [16] A. Wiles: Higher explicit reciprocity laws. Ann. of Math. 107 (1978), no. 2, 235-254. Francesc Bars Cortina, Depart. Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra. Catalonia. Spain. COLEMAN’S POWER SERIES 17 E-mail: francesc@mat.uab.cat Ignazio Longhi, Department of Mathematics, National Taiwan University, No. 1 section 4 Roosevelt Road, Taipei 106, Taiwan. E-mail: longhi@math.ntu.edu.tw