A Note about p-adic Logarithms

Maurizio Candilera

Dipartimento di Matematica Pura e Applicata Via Belzoni, 7 – 35131 Padova - Italy e-mail: candilera@ math.unipd.it

Let k be a perfect field of characteristic p > 0 and denote by W (k) (resp. K_{0}) the ring of Witt vectors over k (resp. its field of fractions). In a joint paper with Valentino Cristante (cf. [PD]), we introduced a ring, denoted by BivR (cf. § 1), in order to compute periods for differentials of the second kind on an abelian variety with good reduction, defined over K_{0}. One of the main features of BivR is that it allows one to define a “good” logarithm, which, in particular, permits a realization of the Tate Module of the multiplicative group over K_{0} as a submodule of the additive group of BivR. In this paper we generalize the constructions given in [PD], replacing the base field K_{0} with a more general local field K and we extend the logarithm map. This generalization appears to be more suitable for treating the case of abelian varieties having semi-stable reduction.

The rings defined here have deep relations with the rings introduced by Fontaine for the study of different types of Galois representations, so a detailed comparison is given.

Here is a list of the content of the paper. In § 1 we extend the definition of BivR and the results of [PD] to a more general base field K and we introduce the ring 𝔹, which appears to be necessary to deal with the extension of the logarithm. This extended map is the content of § 2 and we note here that the proof of Theorem 2.3 adapts an idea of Fontaine (cf. [F88]) to the present situation. Finally, in § 3 we compare our constructions with Fontaine’s rings, extending some results contained in [PD]. In an Appendix we prove that the elements of BivR invariants under the Frobenius map are exactly the elements of ℚ_{p}.

1. Definitions of the basic rings

Let K be a field of characteristic 0, complete with respect to a discrete valuation v, and denote by O_{K} its ring of integers and by k its residue field, perfect, with characteristic p > 0. Denote by W the ring W (k) of Witt vectors over k and by K_{0} its field of fractions. Let [K : K_{0}] = e and denote by K̄ (resp. C) an algebraic closure of K (resp. its completion) and let OK̄ (resp. O_{C}) be the corresponding ring of integers. Finally, denote by G_{K} the Galois group Gal(K̄/K), and consider the extension of the valuation v of K to a valuation v_{C} on C, normalized by setting v_{C}(p) = 1.

We start with a ring of characteristic p. Denote by R = R_{O_{C}} the k–algebra

(1.1)  R = \underleftarrow{lim}_{ℕ}(O_{C}/pO_{C} ⟵^{x^{p}} O_{C}/pO_{C} ⟵^{x^{p}} …).

One can represent an element x of R either as a sequence x = (x_{n})_{n∈ℕ} of elements x_{n} ∈ O_{C}/pO_{C}, or as a sequence x = (x̂_{n})_{n∈ℤ}, where x̂_{n} ∈ O_{C} is defined as follows: for n ≥ 0, choose any element of x̃_{n} ∈ O_{C} which projects to x_{n} modulo pO_{C}, and set x̂_{n} = lim_{m→∞} x̃_{m+n}^{p^{m}}; then, for n < 0, put x̂_{n} = x̂_{0}^{p^{−n}}. One defines a valuation on (the fraction field of) R, by setting v_{R}(x) = v_{C}(x̂_{0}) ∈ ℚ ∪ {∞}. The properties of R can be summarized in the following (cf. [F82] 2.1, or [PD] Proposition 5.6)

1.2 Proposition. Notations as above. The ring R is a perfect k-algebra. Moreover R is a complete valuation ring w.r.t v_{R}, having the algebraic closure k̄ of k as residue field. Finally, there is a natural action of G_{K} on R, obtained by applying the elements of G_{K} to the entries of the sequences (x̂_{n})_{n∈ℤ} ∈ R.

Starting with the perfect ring R, one can construct the ring W (R) of Witt vectors over R and one can put a valuation w on its field of fractions by setting

(1.3) w(ξ) = min { p^{−i}v_{R}(x_{i}) + i : i ≥ 0 } for any ξ = (x_{0}, x_{1}, …) ≠ 0, ξ ∈ W (R)

(cf. [PD] Proposition 5.15 or the following Proposition 1.5 for a slight generalization). We observe that W (R) is a complete subring of the valuation ring and that there is a natural (continuous) action of G_{K} on W (R), induced by the action on R.

One can represent a vector ξ = (x_{0}, x_{1}, …) ∈ W (R) by the (convergent) series ξ = ∑_{n≥0} p^{n}[x_{n}^{p^{−n}}], where, for any x ∈ R, its Teichmüller representative in W (R) is denoted by [x] = (x, 0, 0, …). In what follows, we extend both the representations of vectors to the elements of W (R)[\frac{1}{p}], which will be represented either as a sequence

η = (y_{j})_{j∈ℤ} or as the corresponding (convergent) series η = ∑_{n∈ℤ} p^{n}[y_{n}^{p^{−n}}].

Now take the ring O_{K} and consider W_{K}(R) := W (R) ⊗_{W} O_{K}. Denoting by π any uniformizing parameter of O_{K}, one can represent (uniquely) any element ξ of W_{K}(R) in the form ξ = ∑_{n≥0} π^{n}[x_{n}], with x_{n} ∈ R. In particular, the above sum converges for the tensor product topology and the ring W_{K}(R) is complete for that topology. There is a generalization of the usual Witt polynomials (the theory of “ramified Witt vectors”), which relates the above representation of a sum (resp. product) with the representations of the summands (resp. factors). We summarize these facts in the following Proposition and we refer to [FV], Ch. 1, § 7 or to [Haz] § 6, for a proof.

1.4 Proposition. Notations as above. Let ξ = ∑_{n≥0} π^{n}[x_{n}^{p^{−n}}] and η = ∑_{n≥0} π^{n}[y_{n}^{p^{−n}}] be elements of W_{K}(R) and let ξ + η = ∑_{n≥0} π^{n}[z(+)_{n}^{p^{−n}}] and ξη = ∑_{n≥0} π^{n}[z(×)_{n}^{p^{−n}}].

Then, there exist polynomials S_{n}(X_{0}, …, X_{n}; Y_{0}, …, Y_{n}) and P_{n}(X_{0}, …, X_{n}; Y_{0}, …, Y_{n}) with coefficients in O_{K} such that, for any n ≥ 0, one has

[z(+)_{n}] ≡ S_{n}([x_{0}], …, [x_{n}]; [y_{0}], …, [y_{n}]) mod πW_{K}(R),
[z(×)_{n}] ≡ P_{n}([x_{0}], …, [x_{n}]; [y_{0}], …, [y_{n}]) mod πW_{K}(R).

If, for any i ≥ 0, one sets weight p^{i} to the variables X_{i} and Y_{i}, then S_{n}(X; Y) is an isobaric polynomial with weight p^{n}; and P_{n}(X; Y) is isobaric with weight p^{n} with respect to each group of variables.

The above proposition allows us to extend to W_{K}(R) some of the constructions made on the ring W (R). As a first consequence, we extend the valuation w as follows.

1.5 Proposition. For any ζ = ∑_{n≥0} π^{n}[z_{n}] ∈ W_{K}(R), ζ ≠ 0, set

w_{K}(ζ) = min { v_{R}(z_{i}) + i/e : i ≥ 0 } .

In this way one defines a valuation on the fraction field of W_{K}(R). In particular, the embedding of W (R) into W_{K}(R) is isometric and the topology induced by the valuation w_{K} on W_{K}(R) is equivalent to the tensor product topology.

Proof We maintain the notations of the preceding Proposition. If ξ and η are the elements of W_{K}(R) described in 1.4 and w_{K}(ξ) = a ≤ w_{K}(η) = b; then, for each n ≥ 0, one has v_{R}(x_{n}) ≥ p^{n}(a − n/e) and v_{R}(y_{n}) ≥ p^{n}(b − n/e) ≥ p^{n}(a − n/e). From the congruence in Proposition 1.4 and the isobaricity of polynomials S_{n}(X; Y), one deduces that v_{R}(z(+)_{n}) ≥ p^{n}(a − n/e), for each n, and this means w_{K}(ξ + η) ≥ a. If, moreover, a < b and i_{0} is the least index such that v_{R}(x_{i_{0}}) = p^{i_{0}}(a − i_{0}/e); then

S_{i_{0}}([x_{0}], …, [x_{i_{0}}]; [y_{0}], …, [y_{i_{0}}]) = [x_{i_{0}} + y_{i_{0}}] + ( \begin{matrix}terms of value \\ >p^{i_{0}} (a−i_{0}/e) \end{matrix} ) .

This suffices to conclude that w_{K}(ξ + η) = a.

Now suppose w_{K}(ξ) = a and w_{K}(η) = b and let i_{0} (resp. j_{0}) be the least index such that v_{R}(x_{i_{0}}) = p^{i_{0}}(a − i_{0}/e) (resp. v_{R}(y_{j_{0}}) = p^{j_{0}}(b − j_{0}/e)). The hypothesis on the isobaricity of P_{n}(X; Y) with respect to both variables implies that w_{K}(ξη) ≥ a + b, and the remark that

P_{i_{0}+j_{0}}([x_{0}], …, [x_{i_{0}+j_{0}}]; [y_{0}], …, [y_{i_{0}+j_{0}}]) = [x_{i_{0}}^{p^{j_{0}}} y_{j_{0}}^{p^{i_{0}}} ] + ( \begin{matrix}terms of value \\ >p^{i_{0}+j_{0}} (a+b−(i_{0}+j_{0})/e) \end{matrix} )

allows us to conclude that w_{K}(ξη) = a + b.

The isometricity of the embedding of W (R) into W_{K}(R) follows easily from the remark that p = π^{e}a for some a ∈ O_{K}^{×}. Finally, denoting by W_{r}^{0} the neighborhood of 0 in W (R) made by the vectors having value greater than r, and by W_{r}^{K} the neighborhood of 0 in W_{K}(R) made by the elements having value greater than r, one has

W_{n/e+r}^{K} ⊆ W_{r}^{0} ⊗ O_{K} + W (R) ⊗ π^{n}O_{K} ⊆ W_{r′}^{K},

where r′ = min{r, n/e}. QED

Denote by BivR the closure of the ring W (R)[\frac{1}{p}] in the completion of the fraction field Fr W (R) with respect to the topology induced by the valuation w. There are continuous endomorphisms F (Frobenius) and V (Verschiebung) of BivR, obtained extending by continuity the maps of W (R)[\frac{1}{p}], defined by

(1.6) F ((y_{j})_{j∈ℤ}) = (y_{j}^{p})_{j∈ℤ} = ∑_{n∈ℤ} p^{n}[y_{n}^{p^{−n+1}}]  and  V ((y_{j})_{j∈ℤ}) = ∑_{n∈ℤ} p^{n}[y_{n−1}^{p^{−n}}].

These maps satisfy the relations F V = V F = p. Then, denote by Θ : BivR → C the continuous and surjective homomorphism, which extends by continuity the map Θ : W (R)[\frac{1}{p}] → C, defined by

(1.7)  Θ(ξ) = ∑_{n∈ℤ} p^{n}x̂_{n,n},  for any ξ = (x_{j})_{j∈ℤ} ∈ W (R)[\frac{1}{p}].

Observing that O_{K} is a free W -module of finite rank, one deduces that the ring Biv_{K}R := BivR ⊗_{K_{0}} K = BivR ⊗_{W} O_{K} is the completion of W_{K}(R)[\frac{1}{p}] = W (R)[\frac{1}{p}] ⊗_{W} O_{K} with respect to the topology of the valuation. One can define a continuous and surjective homomorphism Θ_{K} : Biv_{K}R → C, extending the above defined map Θ by K-linearity; in particular this map commutes with the actions of G_{K}.

We summarize some properties of the ring Biv_{K}R and the map Θ_{K} in the following

1.8 Theorem. Notations as above.

(a) The ring Biv_{K}R is a complete topological K–algebra, endowed with a continuous action of the Galois group G_{K}.

(b) The map Θ_{K} is a continuous and surjective ring–homomorphism, equivariant with respect to the actions of the Galois group G_{K}.

(c) The ideal kerΘ_{K} ⊂ Biv_{K}R is principal, and an element α = [a_{0}] + π[a_{1}] + ⋯ ∈ kerΘ_{K} ∩W_{K}(R) is a generator for kerΘ_{K} if, and only if, v_{R}(a_{0}) = 1/e and v_{R}(a_{1}) = 0.

(d) ⋂_{n≥1}(kerΘ_{K})^{n} = {0}.

Proof (sketch). All these results have been proved for the ring BivR (cf. [PD], Theorem 6.10), by means of the valuation and of the properties of Witt polynomials. One can easily adapt those arguments to the present situation. For instance, we sketch the proof of (c). By a direct calculation one can prove that an element α as above is a generator of the kernel of the restriction of Θ_{K} to W_{K}(R) and that this restriction maps W_{K}(R) onto O_{C} (cf. [PD] Proposition 5.8.(b) or [F82] Proposition 2.4). In particular, such element α is again a generator of the kernel of the restriction of Θ_{K} to W_{K}(R)[\frac{1}{p}]. From the surjectivity of the restriction of Θ_{K} to W_{K}(R), it follows that, for any element η ∈ kerΘ_{K}, there exists a sequence (ζ_{n})_{n∈ℕ}, with ζ_{n} ∈ W_{K}(R)[\frac{1}{p}] ∩ kerΘ_{K}, which converges to η. This means that, for each n, one has ζ_{n} = αξ_{n}, for some ξ_{n} ∈ W_{K}(R)[\frac{1}{p}], and we observe that the sequence (ξ_{n})_{n∈ℕ} converges, because w_{K}(ξ_{n} − ξ_{n+1}) = w_{K}(ζ_{n} − ζ_{n+1}) − 1/e for each n. QED

The following Lemma gives an explicit relation between kerΘ_{K} and the kernel of the restriction of Θ_{K} to BivR.

1.9 Lemma. There exists a generator ζ of kerΘ = kerΘ_{K} ∩ W (R) which does not belongs to (kerΘ_{K})^{2}.

Proof Let π be a uniformizing parameter of O_{K}, and let π′ be an element of R such that π̂′_{0} = −π. Denote by f(X) the minimal polynomial for −π over K_{0}. It is well known that f is an Eisenstein polynomial of degree e = [K : K_{0}]. This implies that f([π′]) belongs to kerΘ_{K} ∩ W (R) and that

ζ = f([π′]) ≡ [π′^{e}] + p[a] mod p^{2}W (R), with v_{R}(a) = 0.

Then ζ generates kerΘ (cf. Theorem 1.8.(c) for K = K_{0}). Moreover, in O_{K}[X] one has the decomposition f(X) = (X + π)g(X) with g(−π) ≠ 0, hence in W_{K}(R) we can write ζ = ([π′] + π)g([π′]), where [π′] + π is a generator of kerΘ_{K} (cf. Theorem 1.8.(c)) and Θ_{K}(g([π′])) = g(−π) ≠ 0. This concludes the proof. QED

Now we introduce a sub–W (R)–module bivR of BivR, defined as

bivR = { ∑_{n∈ℤ} p^{n}[x_{n}] : lim inf_{n→+∞} p^{−n}v_{R}(x_{−n}) > 0 } .  (1.10)

Following Barsotti’s [MA], the elements of bivR are called bivectors. If ξ = ∑_{n∈ℤ} p^{n}[x_{n}] is a bivector, then the series defining ξ converges in BivR and in this way one defines an immersion of bivR in BivR. In particular W (R)[\frac{1}{p}] ⊂ bivR and we can extend both the representations introduced for the elements of W (R)[\frac{1}{p}] to the bivectors, so that we can

write ξ = ∑_{n∈ℤ} p^{n}[x_{n}] = (x′_{j})_{j∈ℤ} to denote a bivector, where x′_{j} = x_{j}^{p^{j}} for any j ∈ ℤ. Moreover bivR is closed for the endomorphisms F and V defined in (1.6).

We introduce a logarithm map for the units of W (R) by means of the usual logarithmic series and we quote the following result (cf. [PD] Theorem 7.1), which says, in particular, that the values of this map lie in the submodule bivR.

1.11 Theorem. Let U_{W (R)} = { ξ ∈ W (R) : w(ξ − 1) > 0 }. The map log: U_{W (R)} → BivR is defined by setting

log(ξ) = − ∑_{n≥1} \frac{(1 − ξ)^{n}}{n} .

The above map is a continuous and injective group–homomorphism, with values in the submodule bivR. In particular, it commutes with the actions of the Galois group G_{K} and there is a commutative diagram of G_{K}–equivariant homomorphisms

U_{W (R)} ⟶^{log} bivR
↓^{Θ}  ↓^{Θ}
U_{O_{C}}  ⟶^{log} C

where U_{O_{C}} = { x ∈ O_{C} : v(x − 1) > 0 } and the vertical arrows are the restrictions of the map Θ (cf. (1.7)).

Of course, one can extend the definition of the above map to the multiplicative group U_{W_{K}(R)} = { ξ ∈ W_{K}(R) : w_{K}(ξ − 1) > 0 } and we observe that its values are contained in bivR + W_{K}(R)[\frac{1}{p}], as one can deduce easily by remarking that for any ξ ∈ U_{W_{K}(R)} one has ξ = [x] + πζ with [x] ∈ U_{W (R)} and ζ ∈ W_{K}(R), and that, for n > 0, in K[X, Y] the congruence

\frac{1}{n}(X + πY)^{n} ≡ \frac{X^{n}}{n} mod π^{s}O_{K}[X, Y]

holds, where s = s_{e} := min { p^{r} − re : r ≥ 0 }. In what follows, we are mainly interested in the logarithm of the vectors [u], with u ∈ U_{R} = { x ∈ R : v_{R}(x − 1) > 0 }, so that the map of the above theorem suffices for our purposes.

We can prove some other results about the structure of the ring Biv_{K}R. We begin by introducing an element of kerΘ_{K}. Let ε be an element of R such that ε̂_{0} = 1 ≠ ε̂_{1} and consider [ε] ∈ W (R). One can prove that w(1 − [ε]) = \frac{p}{p−1} so that there exists t = log[ε] ∈ bivR ∩ kerΘ_{K}. In particular, the sub–ℤ_{p}–module spanned by t is isomorphic (as ℤ_{p}[G_{K}]–module) to ℤ_{p}(1), the Tate Module of the multiplicative group 𝔾_{m} (over K). This means that for any g ∈ G_{K}, one has g(t) = χ(g)t, where χ : G_{K} → ℤ_{p}^{×} is the cyclotomic character. As usual, we denote by ℤ_{p}(n) the ℤ_{p}[G_{K}]–module obtained by raising ℤ_{p}(1) to the n^{th} tensor power.

1.12 Proposition. Notations as above.

(a) The element t = log[ε] ∈ BivR belongs to kerΘ_{K} and its image spans the C–vector space kerΘ/(kerΘ)^{2} ≅ kerΘ_{K}/(kerΘ_{K})^{2}.

(b) For any integer n ≥ 0 there is a homomorphism ϕ_{n}: (kerΘ_{K})^{n} → C, which induces an isomorphism of G_{K}–modules ϕ_{n}: (kerΘ_{K})^{n}/(kerΘ_{K})^{n+1} → C(n), where C(n) = C ⊗ ℤ_{p}(n).

(c) For any integer n ≥ 0, one has B(n) = { ξ ∈ Biv_{K}R : g(ξ) = χ(g)^{n}ξ ∀g ∈ G_{K} } = t^{n}K.

Proof It is a consequence of Lemma 1.9 that kerΘ/(kerΘ)^{2} ≅ kerΘ_{K}/(kerΘ_{K})^{2}. By means of that lemma, one can reduce the proof to the case K = K_{0} and for that case, we refer to [PD], Propositions 7.9, 7.11 and Corollary 7.10. QED

The final part of this section is devoted to the description of a new ring. Denote by 𝔹_{K} the completion of Biv_{K}R with respect to the ideal kerΘ_{K} i.e.

𝔹_{K} = \underleftarrow{lim}_{n∈ℕ} Biv_{K}R/(kerΘ_{K})^{n}.  (1.13)

We endow 𝔹_{K} with the projective-limit-topology and consider Biv_{K}R as a subring of 𝔹_{K} (cf. Theorem 1.8.(d)). In particular, we denote by 𝔹 the ring 𝔹_{K_{0}}.

Our first task is to give a description of the above ring and to show that its definition depends only on C and not on K.

1.14 Proposition. The ring 𝔹_{K} is a discrete valuation ring, having C as its residue field and its maximal ideal is spanned by any element of kerΘ_{K} \ (kerΘ_{K})^{2}. Moreover, there is a natural action of G_{K} on 𝔹_{K} and the inclusion BivR → Biv_{K}R induces an isomorphism of rings 𝔹 → 𝔹_{K}.

Proof The proof follows from the facts that θ_{K} is surjective, it commutes with the Galois actions and its kernel is a principal ideal (cf. Theorem 1.8 ). The last claim is a consequence of Lemma 1.9 because the rings 𝔹 and 𝔹_{K} have the same residue field and one can choose the image of the vector ζ of that lemma as uniformizing parameter for both. QED

We remark here that, as a consequence of the above isomorphism, one deduces an embedding of the field K in 𝔹, for any local field K having k as residue field.

We again denote by Θ : 𝔹 → C the map induced by Θ : BivR → C (cf. (1.7)), but, for n > 0, we will use the symbol Fil^{n} to denote the ideal (kerΘ)^{n} of 𝔹 and we set Fil^{0} := 𝔹. In the following Proposition, we summarize some properties of the ring 𝔹 and of the map Θ, which follow from analogous properties of Biv_{K}R (cf. Theorem 1.8 and Proposition 1.12 ).

1.15 Proposition. Notations as above.

(a) The map Θ : 𝔹 → C is a continuous, surjective ring–homomorphism, equivariant with respect to the actions of the Galois group G_{K}.

(b) For any integer n ≥ 0 there is a homomorphism ϕ_{n} : Fil^{n} → C, which induces an isomorphism of G_{K}–modules

ϕ_{n} : \begin{matrix}Fil & n \\ Fil & n+1 \end{matrix} → C(n),

where C(n) = C ⊗ ℤ_{p}(n).

(c) For any integer n ≥ 0, one has

B(n) = { ξ ∈ 𝔹 : g(ξ) = χ(g)^{n}ξ ∀g ∈ G_{K} } = t^{n}K.

Proof (a) follows from Theorem 1.8.(b). It follows from Proposition 1.12 that, for any n ≥ 0, the ideal Fil^{n} ⊂ 𝔹 is spanned by the image of t^{n}, where t = log[ε] is the element of BivR defined above. Then, the map ϕ_{n} is defined by setting ϕ_{n}(ξ) = Θ(ζ), for any ξ = t^{n}ζ ∈ Fil^{n}. The proof of (c) is analogous to the proof of Proposition 1.12.(c), and we now give a sketch of the proof for the convenience of the reader. Let’s start with B(0). If g(ξ) = ξ for all g ∈ G_{K}, then Θ(ξ) ∈ C^{G_{K}} = K. Hence, there exists ξ′ ∈ K ⊂ 𝔹 such that ξ − ξ′ = η ∈ Fil^{1} ∩ B(0). This implies ϕ_{1}(η) = ϕ_{1}(g(η)) = χ(g)g(ϕ_{1}(η)), and then ϕ_{1}(η) = 0, because, for any i > 0, one has C(i)^{G_{K}} = (0) (cf. Theorem 2 of [T]). This implies η ∈ Fil^{2} and, by analogous arguments, one deduces that η ∈ Fil^{n} for each n, which implies η = 0 (cf. Theorem 1.8.(d)) and then B(0) = K.

If ξ ∈ B(n), one has ϕ_{j}(ξ) = 0 for j < n and ξ ≡ t^{n}ξ′ mod Fil^{n+1}, for some ξ′ ∈ K. As before we can write ξ − t^{n}ξ′ = η_{n} ∈ Fil^{n+1} ∩ B(n) where η_{n} satisfies to the condition χ(g)^{n}ϕ_{n+1}(η_{n}) = ϕ_{n+1}(g(η_{n})) = χ(g)^{n+1}g(ϕ_{n+1}(η_{n})), for each g ∈ G_{K}, which implies ϕ_{n+1}(η_{n}) = 0 and then η_{n} ∈ Fil^{n+2}. By the same argument used for η_{1} we get η_{n} ∈ Fil^{k} for each k and then η_{n} = 0, which gives B(n) = t^{n}K. QED

2. Logarithms In this section we give an extension of the logarithm defined in Theorem 1.11 . Let’s

recall some standard notations. If Γ is a commutative group, one defines

the Tate Module of Γ  T_{p}(Γ) := Hom_{ℤ}(ℤ[\frac{1}{p}]/ℤ, Γ),
the Tate Space of Γ  V_{(p)}(Γ) := Hom_{ℤ}(ℤ[\frac{1}{p}], Γ).

For any Γ, there is an exact sequence

0 ⟶ T_{p}(Γ) ⟶ V_{(p)}(Γ) ⟶^{π} Γ

and, in particular, if Γ is a p–divisible group, the map π is surjective.

By applying the above definitions to the multiplicative groups O_{C}^{×} and C^{×}, one gets

V_{(p)}(O_{C}^{×}) = R^{×},  V_{(p)}(C^{×}) = (Fr R)^{×},  T_{p}(O_{C}^{×}) = T_{p}(C^{×}) ≅ ε^{ℤ_{p}} = ℤ_{p}(1),

where the element ε ∈ R is defined in the previous section. The relations among these groups can be summarized in the following diagram with exact rows and columns (cf. [F88])

\begin{aligned}& & & & 0 & & 0 \\ & & & & ↓ & & ↓ \\ 0 & ⟶ & ℤ_{p}(1) & ⟶ & R^{×} & ⟶ & O_{C}^{×} & ⟶ & 0 \\ & & ∥ & & ↓ & & ↓ \\ 0 & ⟶ & ℤ_{p}(1) & ⟶ & (Fr R)^{×} & ⟶ & C^{×} & ⟶ & 0 \\ & & & & ↓^{v_{R}} & & ↓^{v_{C}} \\ & & & & ℚ & & ℚ \\ & & & & ↓ & & ↓ \\ & & & & 0 & & 0 \end{aligned} .  (2.1)

Now we prove the following result about the extension of the logarithm.

2.2 Theorem. Let log : C^{×} → C be a determination of the usual p-adic logarithm and let x ↦ log[x] be the injective map U_{R} → bivR defined by means of Theorem 1.11. Then, there is a unique homomorphism log : V_{(p)}(C^{×}) → 𝔹, which extends the above map and which fits into the following commutative diagram of G_{K}–equivariant maps

\begin{matrix}V_{(p)}(C^{×}) & ⟶^{log} & 𝔹 \\ ↓^{π} & & ↓^{Θ} \\ C^{×} & ⟶^{log} & C \end{matrix} .

Proof As a first approach, suppose that x belongs to V_{(p)}(O_{C}^{×}) = R^{×}. We decompose x as the product x = x_{0}u, where x_{0} ∈ k̄ ⊂ R and u ∈ U_{R} and we set log(x) := log[u]. In this way, we define a homomorphism which commutes with Galois actions and has the desired relations with the usual p-adic logarithm of O_{C}^{×}.

Now fix an element π ∈ R such that π̂_{0} = p. Then, for any x ∈ (Fr R)^{×} = V_{(p)}(C^{×}), there exist –and are uniquely determined– two integers r and s, and an element y ∈ R^{×}, such that s > 0, (r, s) = 1 and π^{r}y = x^{s} (in particular one has v_{R}(x) = \frac{r}{s}). This implies that, for any extension of the above logarithm, we must set

log(x) := \frac{r}{s} log(π) + \frac{1}{s} log[y].

We have reduced our problem to defining a value for log(π), such that it satisfies the compatibility conditions stated above. To satisfy the compatibility condition with the p–adic logarithm, we set log(π) = log \frac{[π]}{p} + log p, where the latter summand is the determination of p-adic logarithm, which we can suppose in K ⊂ 𝔹 (cf. Proposition 1.15 ) because the p-adic logarithm commutes with Galois actions. It remains to observe that 1 − \frac{[π]}{p} belongs to kerΘ, which implies that the usual logarithmic series converges in 𝔹 to an element of Fil^{1}; hence, we set

log \frac{[π]}{p} := − ∑_{n≥1} \frac{(1 − [π]/p)^{n}}{n}

and this concludes our proof. QED

We remark here that the series defining u = log \frac{[π]}{p} does not converge in Biv_{K}R, because w_{K}(1−[π]/p) = 0. Moreover, we may prove a stronger result about the relations between the element u and the ring Biv_{K}R.

2.3 Theorem. Notations as above. The element u = − ∑_{n≥1} \frac{(1−[π]/p)^{n}}{n} of 𝔹 is not algebraic over Fr (Biv_{K}R).

Because any ring Biv_{K}R ≅ BivR⊗_{W} O_{K} is a finite module over BivR, it suffices to prove the claim for the ring BivR. The proof will follow from two Lemmas. We first fix here some notations. Let π ∈ R, with π̂_{0} = p and denote by α = p − [π] this generator of the ideal I := kerΘ_{|W (R)} = Fil^{1} ∩W (R); then, set β := α/p ∈ W (R)[\frac{1}{p}]∩Fil^{1}. Moreover, we set I^{n} := Fil^{n} ∩ W (R) for any n ≥ 0, so that I = I^{1}. We start with the following observation, which gives another description of the image of BivR in 𝔹.

2.4 Lemma. Notations as above. Denote by S the subring of 𝔹 consisting of the series ∑_{n≥0} a_{n}β^{n}, with a_{n} ∈ W (R) for each n ≥ 0. Then BivR ⊆ S[\frac{1}{p}].

Proof By definition, one can write any element of BivR in the form ∑_{n≥0} b_{n}, with b_{n} ∈ W (R)[\frac{1}{p}] for each n ≥ 0, and the above sum converges in the topology induced by the valuation w. We observe that, for any positive integer n, one has

W_{n}^{0} := { x ∈ W (R) : w(x) ≥ n } = ∑_{j=0}^{n} p^{j}I^{n−j}

(cf. [PD], Proposition 5.10.(b)) and from this decomposition one deduces

W_{k} := \{ x ∈ W (R)[\frac{1}{p}] : w(x) ≥ k \} = ∑_{j≥0} p^{−j}W_{k+j}^{0} = ∑_{j≥−k} p^{−j}I^{k+j}.

Given the above sum ∑_{n≥0} b_{n}, for any positive integer k, there exists n_{k} such that b_{n} ∈ W_{k} for each n ≥ n_{k} and then, from the above decomposition, one gets

b_{n} = ∑_{j≥−k} p^{−j}a_{n,j+k}α^{k+j},

for n ≥ n_{k}, where a_{n,j+k} ∈ W (R) and there is only a finite number of summands different from 0. By means of β one can write b_{n} = ∑_{h≥0} a′_{n,h}β^{h}, with a′_{n,h} ∈ p^{k}W (R) for each n ≥ n_{k}. This implies that, for any fixed value of h, the series ∑_{n≥0} a′_{n,h} converges to an element a_{h} ∈ W (R)[\frac{1}{p}] (with respect to the p–adic topology) and then ∑_{n≥0} b_{n} = ∑_{h≥0} a_{h}β^{h} which belongs to S[\frac{1}{p}]. QED

2.5 Lemma. Let u = log \frac{\{π\}}{p} in 𝔹; then u ∉ Fr (BivR).

Proof Let α = p − [π] and β = α/p as above; then u = − ∑_{n≥1} \frac{β^{n}}{n}. From Lemma 2.4 , it follows that it suffices to prove that u ∉ Fr S. Now consider the ideals (cf. § 1)

Fil^{j}S := S ∩ Fil^{j} = β^{j}S for j ≥ 1;

and denote by ϕ_{j} : Fil^{j}S → O_{C} the map which sends aβ^{j} ∈ Fil^{j}S to Θ(a). We observe that it suffices to prove that, if 0 ≠ a ∈ S, then au ∉ S; and observing that ⋂_{n≥0} p^{n}S = (0),

we can write a = p^{r}x with r ≥ 0 and x ∈ S \ pS.

If b ∈ W (R) and Θ(b) ∈ pO_{C}, then b ∈ pW (R) + I ⊆ pS (cf. Theorem 1.8.(c)). This implies that, given x = ∑_{n≥0} b_{n}β^{n} ∈ S \ pS, there exists an integer i ≥ 0 such that

Θ(b_{h}) ∈ pO_{C} for h < i and Θ(b_{i}) ∉ pO_{C}.

Fix an integer j such that j > r and p^{j} > i. From the assumption p^{r}xu ∈ S, we obtain a contradiction. In fact, p^{r}xu ∈ S implies that

−p^{j−1}xu = x ( ∑_{m≥1} p^{j−1} \frac{β^{m}}{m} ) ∈ S;

and then one deduces xβ^{p^{j}}/p ∈ S + Fil^{2p^{j}}. Hence, by the hypothesis on i and j, one gets b_{i}β^{p^{j}+i}/p ∈ S + Fil^{i+p^{j}+1}; and then

ϕ_{i+p^{j}}(b_{i} \frac{β^{p^{j} +i}}{p}) ∈ ϕ_{i+p^{j}}(Fil^{i+p^{j}}S) = O_{C}.

But by a direct calculation one has

ϕ_{i+p^{j}}(b_{i} \frac{β^{p^{j} +i}}{p}) = \frac{Θ(b_{i})}{p} ∉ O_{C}

which gives the desired contradiction. QED

Now we can conclude this section with

Proof of Theorem 2.3. In view of the two previous lemmas, it suffices to prove that, if u is algebraic over Fr (BivR), then u ∈ Fr (BivR).

Let c_{d} +c_{d−1}X +⋯+c_{1}X^{d−1} +X^{d} be the minimal polynomial for u over Fr (BivR). Given g ∈ G_{K_{0}}, one has g(\frac{[π]}{p}) = \frac{[π]}{p}[ε]^{η(g)}, where η : G_{K_{0}} → ℤ_{p} is a continuous map and ε ∈ R is defined in § 1. If t = log[ϵ] ∈ BivR, from the above relation, one obtains g(u) = u + η(g)t because the logarithm map commutes with the actions of G_{K_{0}}.

By observing that BivR and its fraction field are closed with respect to the action of G_{K_{0}}, from the relation

g(c_{d}) + g(c_{d−1})(u + η(g)τ) + ⋯ + g(c_{1})(u + η(g)τ)^{d−1} + (u + η(g)τ)^{d} = 0

and the uniqueness of the minimal polynomial, one finds

g(c_{1}) + dη(g)τ = c_{1} for all g ∈ G_{K_{0}}.

Then the element c = c_{1} +du, satisfies g(c) = c for all g ∈ G_{K_{0}}; and so c ∈ 𝔹^{G_{K_{0}}} = K_{0} ⊂ BivR (cf. Proposition 1.15.(c)). This implies that u = \frac{c−c_{1}}{d} ∈ Fr (BivR) and concludes the proof. QED 3. Comparison with Fontaine’s rings

This section is devoted to explaining the relations among the ring BivR, the module bivR and the rings B^{+}, B_{cris}^{+}, B_{st}^{+}, B_{dR}^{+}, B_{HT}^{+} described by Fontaine in [F88]. Some results of this section are contained also in Remark 7.12 of [PD], but we reproduce them here for the sake of completeness. Throughout this section, we suppose that the base field is K_{0}; this makes it easier to read all definitions and notations. Of course, by means of the results of § 1, one can easily work over a more general base field K.

We start with the definition of the rings B_{cris}^{+} and B^{+}. We denote by α ∈ W (R) a generator of the ideal I = kerΘ ∩ W (R) and let A be the subring of W (R)[\frac{1}{p}] generated over W (R) by the elements \frac{α^{n}}{n!} for n ≥ 0. The ring A does not depend on the choice of α and we denote by J the kernel of the map Θ restricted to A. In particular, there is a natural structure of divided powers on the ideal J of A, defined by setting γ_{n}(x) = \frac{x^{n}}{n!} for each x in J. We denote by A_{cris} the completion of A with respect to the p-adic topology. Finally one sets (cf. [F88] 2.3)

B_{cris}^{+} := A_{cris}[\frac{1}{p}].  (3.1)

The ring B_{cris}^{+} is endowed with a Frobenius endomorphism F, obtained on extending by continuity the analogous map of W (R)[\frac{1}{p}] (cf. (1.6)).

Now we recall the definition of the ring B^{+} (cf. [F82] § 4 and [F88] 4.1.4). For any rational number c ≥ 1, denote by J_{c} the principal ideal J_{c} = { x ∈ R : v_{R}(x) ≥ c } and denote by y_{c} one of its generators. Let S_{J_{c}} := W (R)[\frac{[y_{c}]}{p}] ⊆ W (R)[\frac{1}{p}] and denote by Ŝ_{J_{c}} the completion of S_{J_{c}} with respect to the p–adic topology and set B_{J_{c}}^{+} := Ŝ_{J_{c}}[\frac{1}{p}]. It’s easy to prove that for 1 ≤ c ≤ c′, one has B_{J_{c′}}^{+} ⊆ B_{J_{c}}^{+}. Finally, we define

B^{+} = ⋂_{c≥1} B_{J_{c}}^{+}.  (3.2)

We intend to prove the following

3.3 Proposition. Notations as above. Then

(a) B_{cris}^{+} ⊂ BivR.

(b) B^{+} = ⋂_{n≥1} F^{n}(B_{cris}^{+}).

(c) bivR ⊆ B^{+}.

The above inclusions commute with Frobenius maps and with the actions of the Galois group G_{K_{0}}.

Proof (a).Denoting by α a generator of the ideal I of W (R), any element of A_{cris} can be written in the form

∑_{n≥0} a_{n} \frac{α^{n}}{n!} ,

where the sequence of the coefficients a_{n} ∈ W (R) tends p–adically to zero. The above sum converges in BivR because for any generator α of I one has w(α) = 1 (cf. Theorem 1.8.(c)). This implies

w(a_{n} \frac{α^{n}}{n!} ) ≥ w(a_{n}) + n − v_{p}(n!) ≥ w(a_{n}) + n\frac{p−2}{p−1};

which suffices to conclude, because the coefficients a_{n} ∈ W (R) tend p–adically to zero and so they tend to zero in the topology of the valuation w. (b). We start by proving that Ŝ_{J_{p}} ⊂ A_{cris} ⊂ Ŝ_{J_{1}}. In fact, taking the generator α = [π]−p of I, with π ∈ R such that π̂_{0} = p, we observe that such π ∈ R generates J_{1} and then π^{p} generates J_{p}. This implies that S_{J_{p}} is generated over W (R) by the powers of

\frac{[π^{p}]}{p} = \frac{(α + p)^{p}}{p} = \frac{α^{p}}{p} + pλ,

for some λ ∈ W (R). This implies that

(\frac{α^{p}}{p} + pλ)^{n} = ∑_{j=0}^{n} \binom{n}{j}p^{n−2j}λ^{n−j}α^{pj} = ∑_{j=0}^{n} \binom{n}{j}p^{n−2j}(pj)!λ^{n−j} \frac{α^{pj}}{(pj)!};

and, for j = 0, …, n, the coefficient \binom{n}{j}(pj)!p^{n−2j}λ^{n−j} is in W (R) because n − 2j +

v_{p}((pj)!) = n − j + v_{p}(j!) ≥ 0. This means that all the powers of \frac{[π^{p}]}{p} are in A and then Ŝ_{J_{p}} ⊂ A_{cris}.

In order to prove that A_{cris} ⊂ Ŝ_{J_{1}}, it suffices to show that \frac{α^{n}}{n!} ∈ S_{J_{1}} for n ≥ 0. In fact, one has

\frac{α^{n}}{n!} = \frac{([π] − p)^{n}}{n!} = ∑_{j=0}^{n}(−1)^{n−j} \frac{p^{n−j}}{(n − j)!} \frac{[π]^{j}}{j!} = ∑_{j=0}^{n}(−1)^{n−j} \frac{p^{n}}{j! (n − j)!} \frac{[π]^{j}}{p^{j}}

and \frac{p^{n}}{j!(n−j)!} = \frac{p^{n}}{n!} \binom{n}{j} ∈ W (R).

Then, from the inclusions Ŝ_{J_{p}} ⊂ A_{cris} ⊂ Ŝ_{J_{1}}, one deduces B_{J_{p}}^{+} ⊂ B_{cris}^{+} ⊂ B_{J_{1}}^{+}. Denote by F the Frobenius map on W (R)[\frac{1}{p}] and remark that, for any c ≥ 1, one

has F S_{J_{c}} = S_{J_{pc}}, and then F Ŝ_{J_{c}} = Ŝ_{J_{pc}} and F B_{J_{c}}^{+} = B_{J_{pc}}^{+}. The above inclusions and (3.2) allow us to conclude. (c). Fix c ≥ 1. We will prove that for any ζ = ∑_{n∈ℤ} p^{n}[z_{n}^{p^{−n}}] ∈ bivR, there exists an

integer n_{0} such that p^{n_{0}}ζ ∈ Ŝ_{J_{c}}. We observe that ζ = a+∑_{n>0} ^{\frac{[z_{−n}^{p^{n}}]}{p^{n}}} for some a ∈ W (R), so it suffices to prove the claim for ζ − a. By (1.10) there exists a positive real number b such that v_{R}(z_{−m}) ≥ b for each m >> 0. Then there exists a positive integer n_{0} such that

\frac{[z_{−n}^{p^{n}}]}{p^{n−n_{0}}} ∈ S_{J_{c}} for all n > 0. By noting that v_{R}(z_{−n}^{p^{n}}) − (n − n_{0})c ≥ p^{n}b − (n − n_{0})c diverges

with n, one deduces that the sum p^{n_{0}} ∑_{n>0} ^{\frac{[z_{−n}^{p^{n}}]}{p^{n}}} converges for the p–adic topology of Ŝ_{J_{c}}. This concludes the proof of the inclusion of bivR in B_{J_{c}}^{+} for any c ≥ 1, and thus the inclusion of bivR in B^{+}. QED

We recall that the rings B^{+} e B_{cris}^{+} are endowed with decreasing filtrations

Fil^{n}B^{+} := (kerΘ)^{n} ∩ B^{+}  and  Fil^{n}B_{cris}^{+} := (kerΘ)^{n} ∩ B_{cris}^{+},

with n ≥ 0 and kerΘ ⊂ BivR. We recall the definition of the rings B and B_{cris}. Let ε be the element of R described in § 1 and let t = log[ε] ∈ bivR ⊆ B^{+} ⊆ B_{cris}^{+}. By definition, one sets

B = B^{+}[\frac{1}{t}]  and  B_{cris} = B_{cris}^{+}[\frac{1}{t}].

Now we recall the definitions of the rings B_{dR}^{+} and B_{HT}^{+} (cf. [F82], § 2.8 and [F88], § 1.5). One denotes by Θ_{0} the restriction of Θ to W (R) and one sets

B_{dR}^{+} := \underleftarrow{lim}_{n≥1} (\frac{W (R)}{(kerΘ_{0})^{n}} [\frac{1}{p}]) ,  (3.4)

endowed with the projective-limit-topology of the p–adic topologies on the quotients (Fontaine’s canonical topology).

There is a surjective homomorphism Θ : B_{dR}^{+} → C which can be obtained as limit of the surjective homomorphisms \frac{W (R)}{(kerΘ_{0})^{n}} [\frac{1}{p}] → C (cf. Theorem 1.8 ). One defines a decreasing filtration on B_{dR}^{+} by setting

Fil^{n}B_{dR}^{+} := (kerΘ)^{n} for n ≥ 1 and Fil^{0}B_{dR}^{+} := B_{dR}^{+}.

In particular Fil^{n}B_{dR}^{+}/Fil^{n+1}B_{dR}^{+} ≅ C, for each n.

The ring B_{HT}^{+} is the graded ring associated to the above filtration of B_{dR}^{+}, that is

B_{HT}^{+} = ⨁_{n≥0} Fil^{n}B_{dR}^{+}/Fil^{n+1}B_{dR}^{+}.  (3.5)

In particular, B_{HT}^{+} is isomorphic (in a non-canonical way) to the polinomial ring C[s], where one can choose as indeterminate s the image of any element of Fil^{1} \ Fil^{2}. Finally, we recall that B_{dR} is the field of fractions of B_{dR}^{+} and B_{HT} = B_{HT}^{+} [\frac{1}{s}].

3.6 Proposition. Notations as above.

(a) There is a bicontinuous isomorphism between 𝔹 (cf. (2.6)) and B_{dR}^{+}.

(b) The graded ring associated to the filtration

⋯ ⊆ (kerΘ)^{n+1} ⊆ (kerΘ)^{n} ⊆ ⋯ ⊆ kerΘ ⊆ BivR

is isomorphic to B_{HT}^{+} .

Proof (a). From the above definition it follows that B_{dR}^{+} is the completion of W (R)[\frac{1}{p}] with respect to the kernel of the restriction of Θ to W (R)[\frac{1}{p}]. In particular, the inclusion of W (R)[\frac{1}{p}] into BivR induces a continuous injection of B_{dR}^{+} into 𝔹. This injection is continuous because both the p–adic topology and the w–topology on W (R) induce the same topology on the quotients W (R)/(kerΘ_{0})^{n}[\frac{1}{p}]. In order to prove that this map is a bicontinuous isomorphism, it suffices to prove that for any ξ ∈ BivR and any integer n ≥ 0, there exists ζ ∈ W (R)[\frac{1}{p}] such that ξ − ζ ∈ (kerΘ)^{n}. First of all we observe that there exists η ∈ W (R)[\frac{1}{p}] such that w(ξ − η) ≥ n in BivR and then v_{C}(Θ(ξ − η)) ≥ w(ξ − η) ≥ n. Then there exists λ_{0} ∈ W (R) such that Θ(ξ − η) = p^{n}Θ(λ_{0}) = Θ(p^{n}λ_{0}) and this implies ξ − η − p^{n}λ_{0} ∈ kerΘ. This gives ξ − η − p^{n}λ_{0} = αξ_{1}, with w(ξ_{1}) ≥ n − 1 and then by applying the above arguments one determines λ_{1} ∈ W (R) such that Θ(ξ_{1}) = p^{n−1}Θ(λ_{1}) = Θ(p^{n−1}λ_{1}). This shows ξ−η−p^{n}λ_{0}−p^{n−1}αλ_{1} ∈ (kerΘ)^{2}. Following the same path one gets

ξ − (η + p^{n}λ_{0} + p^{n−1}αλ_{1} + ⋯ + α^{n}λ_{n}) ∈ (kerΘ)^{n},

and ζ = η + p^{n}λ_{0} + p^{n−1}αλ_{1} + ⋯ + α^{n}λ_{n} belongs to W (R)[\frac{1}{p}]. This concludes the proof of the first claim and we observe that (b) follows easily from (a). QED

The above results can be summarized by the following chain of inclusions

(3.7)  bivR ⊂ B^{+} ⊂ B_{cris}^{+} ⊂ BivR ⊂ 𝔹 ≅ B_{dR}^{+}.

There remains to describe the relations with the ring B_{st}^{+}, and we start by recalling the definition of B_{st}^{+} (cf. [F88]).

By means of the map defined in Theorem 1.11 we introduced a homomorphism log : V_{(p)}(O_{C}^{×}) → bivR (cf. Theorem 2.2 ) and, by composing this map with the inclusions bivR ⊂ B^{+} ⊂ B_{cris}^{+} (cf. Proposition 3.3 ), we get a homomorphism

λ : V_{(p)}(O_{C}^{×}) → B_{cris}^{+}.

Then one can consider the category of couples (S, λ_{S}), where S is a B_{cris}^{+}–algebra and λ_{S} : V_{(p)}(C^{×}) → S is a homomorphism which extends the λ defined above. In this category the morphisms are B_{cris}^{+}–algebra-homomorphisms satisfying a natural compatibility. The ring B_{st}^{+} (endowed with a map λ′ : V_{(p)}(C^{×}) → B_{st}^{+}) is by definition the initial object of the above category.

We have observed in the previous section that the quotient V_{(p)}(C^{×})/V_{(p)}(O_{C}^{×}) is a 1dimensional ℚ–vector space (cf.diagram (2.1)), so a homomorphism λ_{S} : V_{(p)}(C^{×}) → S, extending λ : V_{(p)}(O_{C}^{×}) → B_{cris}^{+}, is completely determined by its value on any element b ∈ V_{(p)}(C^{×}), which spans the quotient V_{(p)}(C^{×})/V_{(p)}(O_{C}^{×}). This implies that the above category has an initial object, isomorphic to the couple (B_{st}^{+}, λ′), where

B_{st}^{+} ≅ B_{cris}^{+}[X],  and  λ′(b) = X,  (3.8)

and X is an indeterminate over B_{cris}^{+}.

In particular, the algebra B_{st}^{+} is endowed with a natural extension of the Frobenius map of B_{cris}^{+}, given by setting F (X) := pX. Moreover, there is a natural extension to B_{st}^{+} of the action of G_{K_{0}} on B_{cris}^{+}, which makes λ′ an equivariant map. Finally, there is a unique B_{cris}^{+}–linear derivation

N : B_{st}^{+} → B_{st}^{+}, such that N(λ′(b)) = v_{R}(b) for each b ∈ V_{(p)}(C^{×}).  (3.9)

The above map N is called the monodromy operator associated to v_{R}. The fundamental relation between this derivation and the Frobenius map is NF = pF N.

Finally, one sets B_{st} := B_{cris} ⊗_{B_{cris}^{+}} B_{st}^{+}.

In the previous section, we proved that the homomorphism λ : V_{(p)}(O_{C}^{×}) → B_{cris}^{+}, has an extension log : V_{(p)}(C^{×}) → 𝔹, by setting log[π] = u + log p, where π ∈ R and π̂_{0} = p and u = − ∑_{n≥1} \frac{(1−[π]/p)^{n}}{n} ∈ 𝔹 and log p denotes a determination of p–adic logarithm (cf. Theorem 2.2 ).

In particular, we proved in Theorem 2.3 that the above u ∈ 𝔹 is not algebraic over Fr (BivR). Thus we have

3.10 Proposition. Notations as above. The embedding of B_{cris}^{+} in BivR extends to an embedding of B_{st}^{+} in 𝔹, whose image is contained in the polynomial ring BivR[u].

This embedding commutes with the actions of G_{K_{0}} and with the Frobenius maps. Finally, the BivR–linear derivation N : BivR[u] → BivR[u], defined by setting N(u) = 1, extends the monodromy operator of B_{st}^{+}.

As a final remark we illustrate in the diagram below the relations between the couple (B_{st}^{+}, λ′) and the logarithm of bivectors defined in section 2, where the horizontal maps are all inclusions

bivR ⟶ B_{cris}^{+} ⟶ B_{st}^{+} ⟶ BivR[u] ⟶ B_{dR}^{+}
↑^{log}  ↑^{λ}  ↑_{λ′}  ↑^{log}
V_{(p)}(O_{C}^{×})  V_{(p)}(O_{C}^{×}) ⟶ V_{(p)}(C^{×})  V_{(p)}(C^{×})

4. Appendix.

In this section we’ll prove that the elements of BivR which are stable with respect to the Frobenius map are the elements of ℚ_{p} ≅ biv𝔽_{p} ⊂ bivR ⊂ BivR. This improves the result of Proposition 6.16.(b) of [PD], in which an analogous statement was proved for the elements of bivR.

We start by proving two Lemmas.

4.1 Lemma. Let (ξ_{n})_{n∈ℕ} be a Cauchy sequence in W (R)[\frac{1}{p}] and write ξ_{n} = (x_{n,}_{j})_{j∈ℤ} for each n. Then, there exists an integer j_{0} such that v_{R}(x_{n,j}) > 0 for each j < j_{0} and for each n ∈ ℕ.

Proof If we suppose the claim false, then for any j_{0} ∈ ℤ, there exists n ∈ ℕ and j < j_{0} such that v_{R}(x_{n,j}) = 0. In particular, this implies that w(ξ_{n}) ≤ j < j_{0}, and this is a contradiction, because the values of the terms of a Cauchy sequence are bounded below. QED

4.2 Lemma. Hypotheses as above and suppose that lim_{n→∞} ξ_{n} = lim_{n→∞} F (ξ_{n}). Then there exists an integer j_{0} with the following property: for any positive number k, there exists n_{k} ∈ ℕ such that

n ≥ n_{k}, j < j_{0} ⇒ v_{R}(x_{n,j}) > p^{j}(k − j).

Proof Fix j_{0} as in the previous Lemma. For any k > 0, there exists a positive integer n_{k} such that w(F (ξ_{n}) − ξ_{n}) > k for n > n_{k}. Now fix n > n_{k} and i < j_{0} and write F (ξ_{n})−ξ_{n} = (z_{n,}_{j})_{j∈ℤ}. Since ξ_{n} ∈ W (R)[\frac{1}{p}], we may make the inductive hypothesis that v_{R}x_{n,j}) > p^{j}(k − j) for j < i and we must prove that this implies v_{R}(x_{n,i}) > p^{i}(k − i).

From the condition w(F (ξ_{n}) − ξ_{n}) > k, one deduces v_{R}(z_{n,i}) > p^{i}(k − i). By the usual conditions on polynomials defining the operations on Witt vectors, z_{n,i} is a sum of monomials in the components x_{n,j} with j ≤ i of the kind ax_{n,i−m}^{pλ_{0}+μ_{0}} ⋯ x_{n,i}^{pλ_{m}+μ_{m}}, for some m >> 0, with ∑_{j=0}^{m} p^{j}(λ_{j} + μ_{j}) = p^{m} and a ∈ 𝔽_{p}. Any monomial in which x_{n,i} does not appear has value strictly greater than p^{i}(k − i). The above conditions imply either v_{R}(x_{n,i}) > p^{i}(k − i) or v_{R}(x_{n,i}) = v_{R}(x_{n,i}^{p}) = 0, but the latter cannot happen because j < j_{0}. QED

Finally we are able to prove the main result of this section 4.3 Theorem. Let ξ ∈ BivR such that F (ξ) = ξ. Then ξ ∈ ℚ_{p} ⊂ BivR.

Proof Let (ξ_{n})_{n∈ℕ} be a sequence in W (R)[\frac{1}{p}] converging to ξ and fix j_{0} as in Lemma 4.2 .

For any n ∈ ℕ, we can take the decomposition ξ_{n} = ξ_{n}^{(<j_{0})} + ξ_{n}^{(≥j_{0})}, where ξ_{n}^{(<j_{0})} = (u_{n,}_{j})_{j∈ℤ} ∈ W (R)[\frac{1}{p}] (resp. ξ_{n}^{(≥j_{0})} = (v_{n,}_{j})_{j∈ℤ}) satisfies the conditions u_{n,j} = x_{n,j} (resp. v_{n,j} = 0) for j < j_{0} and u_{n,j} = 0 (resp. v_{n,j} = x_{n,j}) for j ≥ j_{0}.

By Lemma 4.2 , the sequence (ξ_{n}^{(<j_{0})})_{n∈ℕ} converges to zero and so ξ = lim_{n→∞} ξ_{n} =

lim_{n→∞} ξ_{n}^{(≥j_{0})}. As W (R) is complete for the topology of the valuation, this implies that

ξ ∈ W (R)[\frac{1}{p}] and the condition F (ξ) = ξ means that all its components are roots of the polynomial X^{p} − X. This allows us to conclude that the components of ξ belong to the copy of 𝔽_{p} in R, so that ξ ∈ biv𝔽_{p} ≅ ℚ_{p}. QED

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