Exercise 13, § 2

\* We keep the general hypotheses and the notations of Exerc. 12.

a) Let $L$ be a free a-module submodule of $E$ of dimension $n = \dim E$. For every $x \in E - \{0\}$, the set of $a \in A$ such that $ax \in L$ is a fractional ideal of $K$ of the form $m^h$ ($h$ a positive or negative integer); if one sets $\alpha(x) = q^h$, and $\alpha(0) = 0$, now prove that $\alpha$ is an ultranorm on $E$, said to be associated with the free a-module $L$.

b) Conversely, if $\alpha$ is an ultranorm on $E$, the set $L_\alpha$ of the $x \in E$ such that $\alpha(x) \leq 1$ is a free a-module of dimension $n$. If $[\alpha]$ is the norm associated with $L_\alpha$, one has $\alpha \leq [\alpha] \leq q \alpha$, and $[\alpha]$ is the greatest lower bound of the norms associated with free a-modules and which are $\geq \alpha$. One has $[q \alpha] = q.[\alpha]$, and $\alpha(x) = \inf(q^{-t}[q^t \alpha](x))$ for all $x \in E$, where $t$ varies in the interval $[0, 1]$. Moreover, the function $t \mapsto [q^t \alpha](x)$ is left continuous in this interval.

c) With the same notations, now prove that for $0 \leq t \leq 1$, there are at most $n$ distinct ultranorms among the $[q^t \alpha]$. Conversely, let $L$ be the set of ultranorms associated with free a-modules of dimension $n$, and let $(\alpha_t)_{0 \leq t \leq 1}$ be a growing family of ultranorms of $L$ such that $\alpha_1 = q \alpha_0$. Now prove that there exists a basis of $E$ with respect to which all the $\alpha_t$ have the standard form (if $u \in A$ is an element of valuation 1, and $L_t$ the free a-module of the $x \in E$ such that $\alpha_t(x) \leq 1$, consider the vector spaces $L_t/uL_0$ over $k$). Deduce that if, moreover, for every $x \in E$, $t \mapsto \alpha_t(x)$ is left continuous in $[0, 1]$, then there exists one and only one ultranorm $\alpha$ such that $\alpha_t = [q^t \alpha]$ for all $t \in [0, 1]$.

d) The linear group $GL(E)$ operates continuously in $\mathfrak{N}_0(E)$; now prove that it operates properly. For every $\alpha \in \mathfrak{N}_0(E)$, the stabilizer $S_\alpha$ of $\alpha$ in $GL(E)$ is the intersection of the stabilizers of the $[q^t \alpha]$ for $0 \leq t \leq 1$; deduce that $S_\alpha$ is an open and compact subgroup of $GL(E)$, and consequently that the orbit of every $\alpha \in \mathfrak{N}_0(E)$ is a closed discrete subspace of $\mathfrak{N}_0(E)$.

e) For every ultranorm $\alpha \in \mathfrak{N}_0(E)$, consider the decreasing sequence of the dimensions of the $k$-vector spaces $L_t/uL_0$, where $L_t$ is the A-module of the $x \in E$ such that $[q^t \alpha](x) \leq 1$, and $t$ varies from 0 to 1; this sequence is called the sequence of invariants of $\alpha$. For $\alpha$ and $\beta$ to belong to the same orbit in $\mathfrak{N}_0(E)$, it is necessary and sufficient that $X_\alpha = X_\beta$ (Exercise 12, e)) and that the sequences of invariants of $\alpha$ and $\beta$ be the same (use Exercise 12, b)).

f) Deduce from e) that the space of orbits $\mathfrak{N}_0(E)/GL(E)$ is isomorphic to the space of orbits $T^n/\mathscr{S}_n$, where the symmetric group operates on the right on $T^n$ by $(z_1, \ldots, z_n) \mapsto (z_{\sigma(1)}, \ldots, z_{\sigma(n)})$. \*

1 For Exercises 12 and 13, see O. Goldman and N. Iwahori, The space of $p$-adic norms, Acta math., t. CIX (1963), pp. 137-177.