§ 8. Convergence of the Hausdorff series (ultrametric case)

In this paragraph we assume that K is a non-discrete complete valued field of characteristic zero, with an ultrametric absolute value. We denote by $p$ the characteristic of the residue field of K (Commutative Algebra, Chapter VI, § 3, no. 2). If $p \neq 0$ , we write $a = |p|$ ; we know (Commutative Algebra, Chapter VI, § 6, nos. 2 and 3) that $0 < a < 1$ and that there exists one and only one valuation $v$ on K with values in $\mathbf{R}$ whose restriction to $\mathbf{Q}$ is the $p$ -adic valuation $v_p$ and which is such that $|x| = a^{v(x)}$ for all $x \in K$ . Also we write: (1) $\theta = \frac{1}{p-1}.$

If $p = 0$ , we denote by $a$ a real number such that $0 < a < 1$ and by $v$ a valuation on $\mathbf{K}$ with values in $\mathbf{R}$ such that $|x| = a^{v(x)}$ for all $x \in \mathbf{K}$ (loc. cit.). Then $v(x) = 0$ for $x \in \mathbf{Q}^*$ . Also we write:

\theta = 0.

1. $p$ -adic upper bounds of the series exp, log and $\mathbf{H}$

In this no. we assume that $p \neq 0$ .

Lemma 1

Let $n$ be an integer $\geqslant 0$ and let $n = n_0 + n_1 p + \cdots + n_k p^k$ , with $0 \leqslant n_i \leqslant p - 1$ , be the $p$ -adic expansion of $n$ . Let $S(n) = n_0 + n_1 + \cdots + n_k$ . Then*

v_p(n!) = \frac{n - S(n)}{p - 1}.

v_p(n!) = \sum_{i=1}^n v_p(i)

and the number of integers

i

between 1 and

n

for which

v_p(i) \geqslant j

is equal to the integral part

[n/p^j]

n/p^j

. Then

v_p(n!) = \sum_{j \geqslant 0} j ([n/p^j] - [n/p^{j+1}]) = \sum_{j \geqslant 1} [n/p^j].

[n/p^j] = \sum_{i \geqslant j} n_i p^{i-j}

, the lemma follows.

Cited by 2

Cited by Lemma 2, Proposition 14, III § 7

Lemma 2

$v(n) \leqslant v(n!) \leqslant (n - 1)\theta$ and $v(n) \leqslant (\log n)/(\log p)$ for every integer $n \geqslant 1$ . $v(n!) = v_p(n!) = (n - S(n))\theta \leqslant (n - 1)\theta$ by Lemma 1. On the other hand, $n \geqslant p^{v(n)}$ , whence $v(n) \leqslant (\log n)/(\log p)$ . Let $I = \{ U, V \}$ be a set of two elements and let

\mathbf{H} = \sum_{r,s \geqslant 0} H_{r,s}(U, V) \in \hat{\mathbf{L}}_{\mathbf{Q}}(I)

be the Hausdorff series (§ 6, no. 4, Definition 1). Let

\mathbf{Z}_{(p)}

be the local ring of

\mathbf{Z}

relative to the prime ideal

(p)

and

(e_b)_{b \in B}

a basis of

\mathbf{L}_{\mathbf{Z}_{(p)}}(I)

over

\mathbf{Z}

(§ 2, no. 11, Theorem 1). It is also a basis of

\mathbf{L}_{\mathbf{Q}}(I)

over

\mathbf{Q}

Cites 3 · Cited by 3

Cites Lemma 1, Theorem 1, II § 2, Definition 1, II § 6

Cited by § 8. Convergence of the Hausdorff series (ultrametric case), Exercise 1, § 8, Proposition 1

Proposition 1

Let $r$ and $s$ be two integers $\geqslant 0$ . If $H_{r,s} = \sum_{b \in B} \lambda_b e_b$ , where $\lambda_b \in \mathbf{Q}$ , is the decomposition of $\mathbf{H}$ with respect to the basis $(e_b)_{b \in B}$ , then*

v_p(\lambda_b) \geqslant -(r + s - 1)\theta \quad \text{for all } b \in B.

The ring

\mathbf{A}_{\mathbf{Z}_{(p)}}(I)

is identified with the sub-

\mathbf{Z}_{(p)}

-module of

\mathbf{A}_{\mathbf{Q}}(I)

generated by the words

w \in \mathrm{Mo}(I)

. As

\mathbf{L}_{\mathbf{Z}_{(p)}}(I)

is a direct factor of

\mathbf{A}_{\mathbf{Z}_{(p)}}(I)

\mathbf{L}_{\mathbf{Z}_{(p)}}(I) = \mathbf{A}_{\mathbf{Z}_{(p)}}(I) \cap \mathbf{L}_{\mathbf{Q}}(I).

Cites 2 · Cited by 3

Cites II § 6. The Hausdorff series, Lemma 2

Cited by Exercise 1, § 8, Exercise 2, § 8, Lemma 3

Let $f$ be the integer such that $f \leq (r + s - 1)\theta < f + 1$ . Relation (4) is equivalent to $v_p(\lambda_b) \geq -f$ for all $b \in B$ , that is $H_{r,s} \in p^{-f}L_{\mathbf{Z}(p)}$ . But this is equivalent also by (5) to $H_{r,s} \in p^{-f}A_{\mathbf{Z}(p)}(I)$ .

By formula (11) of § 6, no. 4, it suffices to show that, for every integer $m \geq 1$ and all integers $r_1, \ldots, r_m, s_1, \ldots, s_m$ such that

r_1 + \cdots + r_m = r, \quad s_1 + \cdots + s_m = s,

r_i + s_i \geq 1 \quad \text{for } 1 \leq i \leq m,

we have

v_p(m!r!\ldots r_m!s_1!\ldots s_m!) \leq f.

But by Lemma 2,

v_p(r_i!s_i!) \leq (r_i + s_i - 1)\theta

and

v_p(m!) \leq v_p(m!) \leq (m - 1)\theta

; the left hand side of (7) is therefore bounded above by

\theta(m - 1 + \sum_{i=1}^m (r_i + s_i - 1)) = \theta(r + s - 1);

as it is an integer, it is

\leq f

, which completes the proof.

2. Normed Lie algebras

Definition 1

A normed Lie algebra over K* is a Lie algebra with a norm such that

\|x + y\| \leq \sup(\|x\|, \|y\|)

\|[x, y]\| \leq \|x\| \cdot \|y\|

for all

x, y

g

Throughout the rest of this paragraph, g denotes a complete normed Lie algebra.

Cites 2

Cites II § 2. Free Lie algebras, II § 7. Convergence of the Hausdorff series (real or complex case)

For every finite set I we define as in § 7, no. 1 a continuous Lie algebra homomorphism $u \mapsto \tilde{u}$ of $\hat{L}(I)$ into $\hat{P}(g^I; g)$ . We see as in § 7 that if $u = \sum_v u_v$ , with $u_v \in L^v(I)$ for $v \in \mathbf{N}^I$ , then $\tilde{u} = \sum_v \tilde{u}_v$ , where $\tilde{u}_v$ is the polynomial mapping $(t_i)_{i \in I} \mapsto u_v((t_i))$ defined in § 2, no. 4. The composition formula (2) of § 7, no. 1, remains valid.

3. Group defined by a complete normed Lie algebra

Let $H = \sum_{r, s \geq 0} H_{r,s} \in \hat{L}(\{U, V\})$ be the Hausdorff series (§ 6, no. 4, Definition 1). We shall show that the corresponding formal power series with continuous components

\tilde{H} = \sum_{r, s \geq 0} \tilde{H}_{r,s} \in \hat{P}(g \times g, g)

is convergent (Differentiable and Analytic Manifolds, R, 4.1.1).

Let $r \geq 0, s \geq 0$ be such that $r + s \neq 0$ and let $\| \tilde{H}_{r,s} \|$ be the norm of the continuous polynomial $\tilde{H}_{r,s}$ (Differentiable and Analytic Manifolds, R, Appendix, no. 2).

Lemma 3

\| \tilde{H}_{r,s} \| \leq a^{-(r+s-1)\theta}.

Cites 2 · Cited by 2

Cites Proposition 1, II § 2. Free Lie algebras

Cited by Proposition 2, Proposition 5

Let B be a Hall set relative to I and let $H_{r,s} = \sum_{b \in B} \lambda_b e_b$ be the decomposition of $H_{r,s}$ with respect to the corresponding basis of $L(\{U, V\})$ . Then

|\lambda_b| \leq a^{-(r+s-1)\theta}.

This is trivial for

p = 0

, for

\lambda_b \in \mathbf{Q}

; and it follows from Proposition 1 of no. 1 for

p \neq 0

Moreover,

\| \tilde{e}_b \| \leq 1 \quad \text{for } b \in B.

We show more generally by induction on

n

that, for every alternant

b

of degree

n

in the two indeterminates U and V (§ 2, no. 6),

\| \tilde{b} \| \leq 1

. If

n = 1

\tilde{b}

is one of the projections of

g \times g

onto

g

and hence is of norm

\leq 1

; if

n > 1

, there exist two alternants

b_1

and

b_2

of degrees

< n

such that

b = [b_1, b_2]

. As the mapping

\gamma : (x, y) \mapsto [x, y]

g \times g

into

g

is bilinear of norm

\leq 1

, we have (Differentiable and Analytic Manifolds, R, Appendix, no. 4)

\| \tilde{b} \| = \| \gamma \circ (\tilde{b}_1, \tilde{b}_2) \| \leq \| \tilde{b}_1 \| \cdot \| \tilde{b}_2 \| \leq 1.

Relations (11) and (12) imply the lemma.

Proposition 2

The formal power series $\tilde{H}$ is a convergent series (Differentiable and Analytic Manifolds, R, 4.1.1). If G is the ball $\{ x \in g \mid \| x \| < a^\theta \}$ , the domain of absolute convergence of $\tilde{H}$ (Differentiable and Analytic Manifolds, R, 4.1.3) contains $G \times G$ .

Cites 1 · Cited by 3

Cites Lemma 3

Cited by Proposition 4, Proposition 5, Proposition 14, III § 7

If $u$ and $v$ are two real numbers $> 0$ such that $u < a^\theta$ and $v < a^\theta$ , then (Lemma 3)

\| \tilde{H}_{r,s} \| u^r v^s \leq a^\theta (u a^{-\theta})^r (v a^{-\theta})^s

and

\| H_{r,s} \| u^r v^s

tends to 0 when

r + s

tends to infinity.

We denote by $h : G \times G \to g$ the analytic function (Differentiable and Analytic Manifolds, R, 4.2.4) defined by $\tilde{H}$ , that is by the formula

h(x, y) = \sum_{r, s \geq 0} \tilde{H}_{r,s}(x, y) = \sum_{r, s \geq 0} H_{r,s}(x, y) \quad \text{for } (x, y) \in G \times G.

This function is called the Hausdorff function of

g

Let $(x, y) \in G \times G$ . Then

\| \tilde{H}_{r,s}(x, y) \| \leq \sup(\| x \|, \| y \|) \\ \| h(x, y) \| \leq \sup(\| x \|, \| y \|).

(17) follows immediately from (16) and (16) is trivial for $r = s = 0$ ; if $r \geq 1$ , then

\| \tilde{H}_{r,s}(x, y) \| \leq \| \tilde{H}_{r,s} \| \| x \|^{r} \| y \|^{s}

\leq \| x \| \left( \frac{\| x \|}{a^{\theta}} \right)^{r-1} \left( \frac{\| y \|}{a^{\theta}} \right)^{s}

\leq \| x \|;

we argue similarly if

s \geq 1

In particular, $\| h(x, y) \| < a^{\theta}$ for $(x, y) \in G \times G$ .

Proposition 3

Let $G$ be the ball $\{ x \in g | \| x \| < a^{\theta} \}$ . The analytic mapping*

h : G \times G \to G

makes $G$ into a group in which $0$ is the identity element and $-x$ is the inverse of $x$ for all $x \in G$ . Moreover, if $R$ is a real number such that $0 < R < a^{\theta}$ , the ball

\{ x \in g | \| x \| < R \}

(resp. $\{ x \in g | \| x \| \leq R \}$ ) is an open subgroup of $G$ .

As $H(U, -U) = 0$ and $H(0, U) = H(U, 0) = U$ , $h(x, -x) = 0$ and

h(0, x) = h(x, 0) = x

for all

x \in G

. It therefore remains to prove the associativity formula

h(h(x, y), z) = h(x, h(y, z)) \quad \text{for } x, y, z \text{ in } G.

H(H(U, V), W) = H(U, H(V, W))

\hat{L}(\{ U, V, W \})

(§ 6, no. 5, Proposition 4), we have

\tilde{H} \circ (\tilde{H} \times \mathrm{Id}_g) = \tilde{H} \circ (\mathrm{Id}_g \times \tilde{H})

\hat{P}(g \times g \times g; g)

(no. 2) and (19) implies (18) by (16) and Differentiable and Analytic Manifolds, R, 4.1.5.

In other words (Chapter III, § 1), $G$ with the Hausdorff function is a Lie group.

Cites 2 · Cited by 1

Cites Proposition 4, II § 6, III § 1. Lie groups

Cited by Lemma 3, III § 4

4. Exponential in complete normed associative algebras

In this no. A will denote a unital associative algebra with a norm $x \mapsto \| x \|$ satisfying the conditions:

\| x + y \| \leq \sup(\| x \|, \| y \|) \\ \| xy \| \leq \| x \| \cdot \| y \| \\ \| 1 \| = 1

for $x, y$ in $A$ , and complete with this norm. The results of the second and third paragraphs of § 7, no. 3, remain valid.

We take $I = \{ U \}$ and consider the images $\tilde{e}$ and $\tilde{l}$ of the series $e(U) = \sum_{n \geq 1} \frac{U^n}{n!}$ and $l(U) = \sum_{n \geq 1} (-1)^{n-1} \frac{U^n}{n}$ in $\hat{P}(A; A)$ . Then:

\left\| \left( \frac{U^n}{n!} \right)^{\sim} \right\| \leq a^{-(n-1)\theta}

\left\| \left( \frac{U^n}{n} \right)^{\sim} \right\| \leq a^{-\frac{\log n}{\log p}}

by Lemma 2 of no. 1. Hence the radius of absolute convergence of the series $\tilde{e}$ (resp. $\tilde{l}$ ) is $\geq a^\theta$ (resp. $\geq 1$ ) (Differentiable and Analytic Manifolds, R, 4.1.3). For $R > 0$ , let $G_R = \{ x \in A \mid \|x\| < R \}$ ; we write $G = G_0$ . The series $\tilde{e}$ (resp. $\tilde{l}$ ) defines an analytic mapping $e_A$ (resp. $l_A$ ) of $G$ (resp. $G_1$ ) into $A$ . We write:

\exp_A(x) = 1 + e_A(x) = \sum_{n \geq 0} \frac{x^n}{n!} \quad \text{for } x \in G

\log_A(x) = l_A(x-1) = \sum_{n \geq 1} (-1)^{n-1} \frac{(x-1)^n}{n} \quad \text{for } x-1 \in G_1'

(we omit the index

A

when no confusion can arise). For

x \in G_R

and

n \geq 1

\left\| \frac{x^n}{n} \right\| \leq \left\| \frac{x^n}{n!} \right\| < R^n a^{-(n-1)\theta} = R \left( \frac{R}{a^\theta} \right)^{n-1}

and hence

e_A(G_R) \subset G_R

l_A(G_R) \subset G_R

for

R \leq a^\theta

Proposition 4

Let $R$ be a real number such that $0 < R \leq a^\theta$ . The mapping $\exp_A$ defines an analytic isomorphism of $G_R$ onto $1 + G_R$ and the inverse isomorphism is the restriction of $\log_A$ to $1 + G_R$ .

Cites 1

Cites Proposition 2

$e(l(X)) = l(e(X)) = X$ . By (20), (21) and Differentiable and Analytic Manifolds, R, 4.1.5, we deduce that $e_A(l_A(x)) = l_A(e_A(x))$ for $x \in G_R$ . Then

\exp_A(\log_A x) = x \quad \text{for } x \in 1 + G_R \\ \log_A(\exp_A x) = x \quad \text{for } x \in G_R

which completes the proof.

If $A$ is given the bracket $[x, y] = xy - yx$ , $A$ becomes a complete normed Lie algebra, for $\|xy - yx\| \leq \sup(\|xy\|, \|yx\|) \leq \|x\| \cdot \|y\|$ . Proposition 2 of no. 3 implies that the domain of absolute convergence of $\tilde{H}$ contains $G \times G$ and $\tilde{H}$ therefore defines an analytic function $h : G \times G \to A$ ; then

h(x, y) = \sum_{r, s \geq 0} H_{r,s}(x, y).

Proposition 5

For x, y in G,

\exp_A \cdot \exp_A y = \exp_A h(x, y).

e^{U e^V} = e^{H(U, V)}

and hence

m \circ (1 + \tilde{\epsilon}, 1 + \tilde{\epsilon}) = (1 + \tilde{\epsilon}) \circ \tilde{H}

\tilde{H}(A \times A; A)

(where m denotes multiplication on A). The proposition then follows from Proposition 2, Lemma 3 and Differentiable and Analytic Manifolds, R, 4.1.5.

Cites 2

Cites Lemma 3, Proposition 2

Exercises

We suppose that the residual characteristic $p$ of the field $K$ is >0. We denote by $\mathfrak{o}_K$ the ring of the valuation $v$ of $K$ .

See the exercises for § 8.