Lubin’s conjecture for full $p$-adic dynamical systems

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P ublications mathématiques de Besançon

A l g è b r e e t t h é o r i e d e s n o m b r e s

Laurent Berger

Lubin’s conjecture for full p-adic dynamical systems

2016, p. 19-24.

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LUBIN’S CONJECTURE FOR FULL p-ADIC DYNAMICAL SYSTEMS by

Laurent Berger

Abstract. — We give a short proof of a conjecture of Lubin concerning certain families of p-adic power series that commute under composition. We prove that if the family isfull (large enough), there exists a Lubin-Tate formal group such that all the power series in the family are endomorphisms of this group. The proof uses ramification theory and somep-adic Hodge theory.

Résumé. — (La conjecture de Lubin pour les systèmes dynamiques p-adiques pleins) Nous donnons une démonstration courte d’une conjecture de Lubin concernant certaines familles de séries formellesp-adiques qui commutent pour la composition. Nous montrons que si la famille estpleine (assez grosse), il existe un groupe formel de Lubin-Tate tel que toutes les séries de la famille sont des endomorphismes de ce groupe. La démonstration utilise la théorie de la ramification et un peu de théorie de Hodgep-adique.

Introduction

Let K be a finite extension of Q_p, and let O_K be its ring of integers. In [5], Lubin studied p-adic dynamical systems, namely families of elements of T · O_K[[T]] that commute under composition, and remarked that “experimental evidence seems to suggest that for an invertible series to commute with a noninvertible series, there must be a formal group somehow in the background”. This observation has motivated the work of a number of people (Hsia, Laubie, Li, Movaheddi, Salinier, Sarkis, Specter, ...) who proved various results in that direction. The purpose of this note is to give a proof of a special case of the above observation, which is referred to as “Lubin’s conjecture” in §3.1 of [7]. Let us consider a family F of commuting power series F(T) ∈ T · O_K[[T]]. We say that such a family is full if for all α ∈ O_K there existsF_α(T)∈ F such thatF_α⁰(0) =α and if wideg(F_π(T)) =q, where wideg(F(T)) denotes the Weierstrass degree of F(T), π is any uniformizer of O_K and q is the cardinality of the residue field of O_K.

Mathematical subject classification (2010). — 11S82, 11S15, 11S20, 11S31, 13F25, 13F35, 14F30.

Key words and phrases. — p-adic dynamical system, Lubin-Tate formal group,p-adic Hodge theory.

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20 Lubin’s conjecture for fullp-adic dynamical systems

Theorem. — If F is a full family of commuting power series, there exists a Lubin-Tate formal group G such that F_α(T)∈End(G) for all α∈ O_K.

This result already appears as Theorem 2 of [4]. Our proof is however considerably shorter than that of ibid., and does not use the theory of the field of norms. It is very similar to that of the main result of [8], which treats the caseK=Q_p. The main ingredients are ramification theory and some p-adic Hodge theory. In order to simplify the use of p-adic Hodge theory, we assume that K is a Galois extension ofQ_p.

1. p-adic dynamical systems

In this note, we consider a setF ={F_α(T)}_α∈O_K of power seriesFα(T)∈T·O_K[[T]] such that F_α⁰(0) =αand F_α◦F_β(T) =F_β◦F_α(T) whenever α,β ∈ O_K. Recall thatπ is a uniformizer of O_K, and that q is the cardinality of the residue field k of O_K. If F(T) is a power series and n>0, we denote byF^◦n(T) then-th fold iteration F◦ · · · ◦F(T). IfF(T) has an inverse for the composition, this definition extends to n ∈ Z. Recall that the Weierstrass degree wideg(F(T)) ofF(T) =^P^+∞_i=1 fiTⁱ ∈T · O_K[[T]] is the smallest integeri such that fi ∈ O^×_K. By the Weierstrass preparation theorem, if wideg(F)6= +∞, then F has wideg(F) zeroes in m_C_p.

Proposition 1.1. — There exists a power seriesG(T)∈T·k[[T]]and an integer d>1such that G⁰(0)∈k^× and Fπ(T)≡G(T^p^d).

Proof. — See (the proof of) theorem 6.3 and corollary 6.2.1 of [5].

Proposition 1.2. — There exists a power series L_F(T)∈K[[T]] such that 1. L_F(T) =T + O(T²);

2. L_F(T) converges on the open unit disk;

3. L_F ◦Fα(T) =α·L_F(T) for all α∈ O_K.

Proof. — See propositions 1.2 and 2.2 of [5] for the construction of a unique power series L_F(T) that satisfies (1), (2) and (3) for α a uniformizer of O_K. If β ∈ O_K \ {0}, then β⁻¹·L_F◦F_β also satisfies (1), (2) and (3) forαas above, so that L_F◦F_β(T) =β·L_F(T) for

all β ∈ O_K.

The hypothesis that F is full implies thatp^d =q, so that wideg(Fπ(T)) =q. For n>1, let Λ_ndenote the set ofu∈m_C_p such thatF_π^◦n(u) = 0 andF_π^◦n−1(u)6= 0 and let Λ_∞=∪_n>1Λ_n. Proposition 1.1 implies that F_π⁰(T)/π is a unit of O_K[[T]], so that the roots of F_π^◦n(T) are simple for all n>1. The set Λ_n therefore hasqⁿ⁻¹(q−1) elements.

The series Fα(T) is invertible ifα∈ O^×_K so that in this case,Fα(z) = 0 if and only ifz= 0. If u∈Λ_nandα∈ O^×_K, thenF_π^◦n◦F_α(u) =F_α◦F_π^◦n(u) = 0 andF_π^◦n−1◦F_α(u) =F_α◦F_π^◦n−1(u)6=

0 so that the action of Fα(T) permutes the elements of Λ_n.

Let Kn =K(Λn), so that Λ_i ⊂Kn if i6n, and let K∞=∪_n>1Kn. If α ∈ O_K^×, let n(α) be the largest integer n>0 such thatα∈1 +πⁿO_K.

Proposition 1.3. — If n>1 and u∈Λ_n, then

Publications mathématiques de Besançon – 2016

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1. Fα(u) =u if and only if n(α)>n;

2. If n(α) =n, then wideg(F_α(T)−T) =qⁿ; 3. Λ_n={F_α(u)}_α∈O×

K

.

Proof. — If n = 1 and F_α(u) = u, then u is a root of F_α(T)−T = (α −1)T + O(T²), so that α−1 ∈ πO_K. This implies that {F_α(u)}_α∈O^×

K

has at least q −1 distinct elements.

These elements are all roots ofF_π(T)/T, whose wideg is q−1, so{F_α(u)}_α∈O× K

has precisely q−1 elements. These elements all have valuation 1/(q−1), and if n(α) = 1, the Newton polygon of F_α(T)−T starts at the point (1,1), so that it can have only one segment, and wideg(Fα(T)−T) =q. This implies the proposition for n= 1.

Assume now that the proposition holds up to some n>1 and take u ∈Λ_n+1. If n(α)6 n, then Fα(T)−T has at most qⁿ roots by (2), contained in Λ₀∪. . .∪Λ_n by (1). Therefore F_α(u) =uimpliesn(α)>n+ 1. The set{F_α(u)}_α∈O×

K

therefore has at leastqⁿ(q−1) distinct elements, all of them roots of F_π^◦n+1(T)/F_π^◦n(T).

This implies that{F_α(u)}_α∈O^×

K has exactlyqⁿ(q−1) elements. If n(α) =n+ 1, the Newton polygon of F_α(T) −T starts at the point (1, n+ 1), with n+ 1 segments of height one and slopes −1/q^k(q−1) with 0 6 k 6 n, so that it reaches the point (qⁿ⁺¹,0) and hence wideg(F_α(T)−T) =qⁿ⁺¹. This implies the proposition forn+ 1.

Corollary 1.4. — The field K∞ is an abelian totally ramified extension of K, and if g ∈ Gal(K∞/K), there is a unique η(g)∈ O^×_K such that g(u) =F_η(g)(u) for all u∈Λ_∞.

The mapη : Gal(K∞/K)→ O_K^× is an isomorphism.

Proof. — Take u ∈ Λ_n and α ∈ O^×_K. As we have seen above, F_α(u) ∈ Λ_n, so that the map u7→F_α(u) induces a field automorphism ofK(u). By (3) of Proposition 1.3, this implies that K_n=K(u) and that every element of Gal(K_n/K) comes fromu7→F_α(u) for some α∈ O_K^×. The extension Kn/K is therefore abelian, and so is K∞/K. SinceKn=K(u), the extension K_n/K is totally ramified, and so isK∞/K.

The mapη is surjective because every F_α(T) gives rise to an automorphism ofK∞, and it is injective because if η(g) = 1, then g(u) =u for all u∈Λ_∞ so that g= 1.

In order to prove our main theorem, we study thep-adic periods ofη. Corollary 1.4 and local class field theory imply that the extensionK∞/K is attached to a uniformizer$ofO_K. Let χ$:G_K → O_K^× denote the corresponding Lubin-Tate character.

2. p-adic Hodge theory

Let R be the p-adic completion of lim−→n>1O_K[[Xn]] where O_K[[Xn]] is seen as a subring of O_K[[Xn+1]] via the identification Xn = Fπ(Xn+1) (this ring is defined in [8], where it is denoted by A∞). We define an action of G_K on R by g(H(X_n)) = H(F_η(g)(X_n)). This is well-defined since F_π◦F_η(g)(T) =F_η(g)◦F_π(T). We haveR/πR= lim

−→n>1k[[X_n]].

Lemma 2.1. — The ring R/πR is perfect.

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Proof. — LetG(T) be as in Lemma 1.1. The fact thatXn=Fπ(Xn+1) implies thatG^◦n(Xn) = G^◦n+1(X_n+1)^q. Since G⁰(0)∈k^×, we havek[[T]] =k[[G(T)]] and therefore

R/πR= lim

G^◦n(Xn)=G−→^◦n+1(Xn+1)^q

k[[G^◦n(X_n)]]

is perfect.

Let Ee⁺ = lim

←−^x7→x^qO_C_p/π. Choose a sequence {u_n}_n>1 with u_n ∈ Λ_n and F_π(u_n+1) = u_n. This sequence gives rise to a map i : R/πR → Ee⁺, determined by the requirement that i(X_n) = (G^◦1(u_n), G^◦2(u_n+1), . . .). The definition of the action ofG_K onR and Corollary 1.4 imply that iisGK-equivariant.

Lemma 2.2. — The map i:R/πR→Ee⁺ is injective.

Proof. — It is enough to show that i : k[[Xn]] → Ee⁺ is injective. If it was not, there would be a nonzero polynomial P(T) ∈ k[T] such that P(i(X_n)) = 0, and then i(X_n) = (G^◦1(u_n), G^◦2(u_n+1), . . .) would belong to F_p, which is clearly not the case.

Let K₀ =Q^unr_p ∩K and let ˜A⁺ =O_K⊗_O_K

0 W(Ee⁺) (see [2]; note that ˜A⁺ usually denotes W(Ee⁺), and is denoted by Ainf in ibid.). We have R = O_K ⊗O_K

0 W(R/πR) since R is a strict π-ring, and by the functoriality of Witt vectors, the map iextends to an injective and G_K-equivariant mapi:R → A˜⁺. We write x instead of i(X₁)∈A˜⁺. The G_K-equivariance of iimplies thatg(x) =F_η(g)(x).

Let B⁺_cris and B_dR be some of Fontaine’s rings of periods. Recall that B_dR is a field, that there is a Frobenius map ϕ on B⁺_cris, a filtration {FilⁱB_dR}_i∈Z on B_dR, and an injective map K ⊗_K₀ B⁺_cris → B_dR. There is also an action of G_K on B⁺_cris and B_dR compatible with the above structure, and B^G_dR^K = K. Let ϕq = ϕ^f on B⁺_cris, where q = p^f, extended by K-linearity to K ⊗_K₀ B⁺_cris. We refer to [2] and [3] for the properties of these objects.

Let Σ = Gal(K/Q_p). If τ ∈ Σ, choose a n(τ) ∈ Z_>0 such that τ|_K₀ = ϕ^n(τ). The map τ ⊗ϕ^n(τ) :K⊗_K₀ B⁺_cris → K⊗_K₀ B⁺_cris is then well-defined and commutes with ϕ_q and the action of G_K.

We say that a character λ : G_K → O_K^× is crystalline positive if there exists a nonzero z∈K⊗_K₀B⁺_crissuch thatg(z) =λ(g)·zfor allg∈G_K. The following proposition summarizes the input that we need from the p-adic Hodge theory of characters.

Proposition 2.3. — A character λ:G_K → O^×_K that factors through Gal(K∞/K) is crys- talline positive if and only if λ=^Q_τ∈Στ ◦χ^h_$^τ withhτ ∈Z_>0.

If t_$ ∈K⊗_K₀ B⁺_cris is such that g(t_$) =χ_$(g)·t_$ for all g∈G_K, then t_$ ∈Fil¹B_dR and ϕq(t$) =$·t$.

Sketch of proof. — If λ:GK→ O_K^× is a crystalline positive character andhτ ∈Z_>0 denotes theHodge-Tate weightofλatτ ∈Σ, thenλ·^Q_τ∈Στ◦χ^−h_$ ^τ is crystalline and has Hodge-Tate weight zero at all τ ∈ Σ so that it is unramified, and therefore trivial if λ factors through Gal(K∞/K), since K∞/K is totally ramified.

Letω_E andt_E be the elements constructed in §9.2 and §9.3 of [1] (withE =K andπ_E =$).

We have t_E ∈ K ⊗_K₀ B⁺_cris and ϕ_q(t_E) = $·t_E and t_E ∈ Fil¹B_dR (proposition 9.10 of ibid). If g ∈ GK, then (in the notation of ibid and where [·]_LT denotes the endomorphisms

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of the Lubin-Tate group attached to $) we have g(ωE) = [χ$(g)]_LT(ωE) and therefore g(t_E) = g(F_E(ω_E)) = F_E(g(ω_E)) = F_E ◦[χ_$(g)]_LT(ω_E) = χ_$(g)·F_E(ω_E) = χ_$(g)·t_E sinceF_E is the logarithm of the Lubin-Tate group attached to$. Ift_$∈K⊗_K₀B⁺_cris is such thatg(t$) =χ$(g)·t$ for all g∈GK, then t$/tE ∈B^G_dR^K =K, and this implies the rest of

the proposition.

Recall that L_F(T) ∈ K[[T]] is the logarithm attached to F. Since L_F(T) converges on the open unit disk, we can view L_F(x) as an element ofK⊗_K₀ B⁺_cris.

Proposition 2.4. — The characterη:G_K → O^×_K is crystalline positive.

Proof. — Ifg∈G_K, theng(LF(x)) = LF(g(x)) = LF(F_η(g)(x)) =η(g)·L_F(x).

Corollary 2.5. — We haveL_F(x) =β·^Q_τ∈Σ(τ⊗ϕ^n(τ))(t$)^h^τ wherehτ ∈Z_>0 andβ ∈K^×. Proof. — This follows from the facts thatη=^Q_τ∈Στ◦χ^h_$^τ, thatχ$(g) =g(t$)/t$and that

B^G_dR^K =K.

Proposition 2.6. — We have ϕ_q(L_F(x)) =µ·L_F(x) for some µ∈πO_K.

Proof. — Corollary 2.5 and Proposition 2.3 imply the proposition with µ=^Q_ττ($)^h^τ, and

not all hτ can be equal to 0 since η6= 1.

Corollary 2.7. — We have ϕq(x) =Fµ(x).

Proof. — Proposition 2.6 implies that L_F(ϕq(x)) = LF(Fµ(x)). We would like to apply L^◦−1_F (T) but we have to mind the convergence and need to proceed as follows. Since η is nontrivial, there is aτ ∈Σ such that h_τ⁻¹ >1. We have

(τ ⊗ϕ^n(τ))(L_F(ϕ_q(x))) = (τ ⊗ϕ^n(τ))(L_F(F_µ(x)))

in K⊗_K₀ B⁺_cris and h_τ⁻¹ > 1 now implies that (τ ⊗ϕ^n(τ))(L_F(ϕq(x))) is divisible by t$ so that by Proposition 2.3, it belongs to Fil¹B_dR. We can then apply L^τ◦−1_F (T) in B_dR and get that (τ⊗ϕ^n(τ))(ϕq(x)) = (τ⊗ϕ^n(τ))(Fµ(x)) inB_dR. This equality also holds in ˜A⁺, so that

ϕ_q(x) =F_µ(x).

Theorem 2.8. — There is a Lubin-Tate formal group Gsuch that F_α(T)∈End(G) for all α∈ O_K.

Proof. — By Corollary 2.7, we have ϕq(x) = Fµ(x). This implies that Fµ(T) ≡ T^q mod πO_K[[T]]. The Weierstrass degree ofF_µ(T) isq^val(µ)so that val(µ) = 1 andF_µ(T) is a Lubin- Tate power series. By [6], there is a Lubin-Tate formal group G such thatF_µ(T)∈End(G).

Since Fα(T) commutes withFµ(T), we also haveFα(T)∈End(G) for all α∈ O_K. Remark 2.9. — We have µ= $ and η =χ$. Indeed, the extension K∞/K is generated by the torsion points ofG, and is therefore attached to µ by local class field theory, so that µ=$. This in turn implies that η=χ$.

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[3] Jean-Marc Fontaine. Représentations p-adiques semi-stables.Astérisque, (223):113–184, 1994.

[4] Liang-Chung Hsia and Hua-Chieh Li. Ramification filtrations of certain abelian Lie extensions of local fields.J. Number Theory, 168:135–153, 2016.

[5] Jonathan Lubin. Nonarchimedean dynamical systems.Compositio Math., 94(3):321–346, 1994.

[6] Jonathan Lubin and John Tate. Formal complex multiplication in local fields.Ann. of Math. (2), 81:380–387, 1965.

[7] Ghassan Sarkis. Height-one commuting power series overZp. Bull. Lond. Math. Soc., 42(3):381–

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March 30, 2016

Laurent Berger, UMPA de l’ENS de Lyon, UMR 5669 du CNRS, IUF E-mail :[email protected]•Url :perso.ens-lyon.fr/laurent.berger/