When Baire meets Krasner

Here is a well-but-ought-to-be-better known theorem.

Theorem. — Let $\ell$ be a prime number and let $G$ be a compact subgroup of $\mathop{\rm GL}_d(\overline{\mathbf Q_\ell})$. Then there exists a finite extension $E$ of $\mathbf Q_\ell$ such that $G$ is contained in $\mathop{\rm GL}_d(E)$.

Before explaining its proof, let us recall why such a theorem can be of any interest at all. The keyword here is Galois representations.

It is now a well-established fact that linear representations are an extremly useful tool to study groups. This is standard for finite groups, for which complex linear representations appear at one point or another of graduate studies, and its topological version is even more classical for the abelian groups $\mathbf R/\mathbf Z$ (Fourier series) and $\mathbf R$ (Fourier integrals). On the other hand, some groups are extremly difficult to grasp while their representations are ubiquitous, namely the absolute Galois groups $G_K=\operatorname{Gal}(\overline K/K)$ of fields $K$.

With the notable exception of real closed fields, these groups are  infinite and have a natural (profinite) topology with open subgroups the groups $\operatorname{Gal}(\overline K/L)$, where $L$ is a finite extension of $K$ lying in $\overline K$. It is therefore important to study their continuous linear representations. Complex representations are important but since $G_K$ is totally discontinuous, their image is always finite. Therefore, $\ell$-adic representations, namely continuous morphisms from $G_K$ to $\mathop{\rm GL}_d(\mathbf Q_\ell)$, are more important. Here $\mathbf Q_\ell$ is the field of $\ell$-adic numbers.

Their use goes back to Weil's proof of the Riemann hypothesis for curves over finite fields, via the action on $\ell^\infty$-division points of its Jacobian variety. Here $\ell$ is a prime different from the characteristic of the ground field. More generally, every Abelian variety $A$ over a field $K$ of characteristic $\neq\ell$ gives rise to a Tate module $T_\ell(A)$ which is a free $\mathbf Z_\ell$-module of rank $d=2\dim(A)$, endowed with a continuous action $\rho_{A,\ell}$ of  $G_K$. Taking a basis of $T_\ell(A)$, one thus has a continuous morphism $G_K\to \mathop{\rm GL}_d(\mathbf Z_\ell)$, and, embedding $\mathbf Z_\ell$ in the field of $\ell$-adic numbers,  a continuous morphism $G_K\to\mathop{\rm GL}_d(\mathbf Q_\ell)$. Even more generally, one can consider the $\ell$-adic étale cohomology of algebraic varieties over $K$.

For various reasons, such as the need to diagonalize additional group actions, one can be led to consider similar representations where $\mathbf Q_\ell$ is replaced by a finite extension of $\mathbf Q_\ell$, or even by the algebraic closure $\overline{\mathbf Q_\ell}$. Since $G_K$ is a compact topological groups, its image by a continuous representation $\rho\colon G_K\to\mathop{\rm GL}_d(\overline{\mathbf Q_\ell}$ is a compact subgroup of $\mathop{\rm GL}_d(\overline{\mathbf Q_\ell}$ to which the above theorem applies.

This being said for the motivation, one proof (attributed to Warren Sinnott)  is given by Keith Conrad in his short note, Compact subgroups of ${\rm GL}_n(\overline{\mathbf Q}_p)$. In fact, while browsing at his large set of excellent expository notes,  I fell on that one and felt urged to write this blog post.

The following proof had been explained to me by Jean-Benoît Bost almost exactly 20 years ago. I believe that it ought to be much more widely known.

It relies on the Baire category theorem and on Krasner's lemma.

Lemma 1 (essentially Baire). — Let $G$ be a compact topological group and let $(G_n)$ be an increasing sequence of closed subgroups of $G$ such that $\bigcup G_n=G$. There exists an integer $n$ such that $G_n=G$.

Proof. Since $G$ is compact Hausdorff, it satisfies the Baire category theorem and there exists an integer $m$ such that $G_m$ contains a non-empty open subset $V$. For every $g\in V$, then $V\cdot g^{-1}$ is an open neighborhood of identity contained in $G_m$. This shows that $G_n$ is open in $G$. Since $G$ is compact, it has finitely many cosets $g_iG_m$ modulo $G_m$; there exists an integer $n\geq m$ such that $g_i\in G_n$ for every $i$, hence $G=G_n$. QED.

Lemma 2 (essentially Krasner). — For every integer $d$, the set of all extensions of $\mathbf Q_\ell$ of degree $d$, contained in $\overline{\mathbf Q_\ell}$, is finite.

Proof. Every finite extension of $\mathbf Q_\ell$ has a primitive element whose minimal polynomial can be taken monic and with coefficients in $\mathbf Z_\ell$; its degree is the degree of the polynomial. On the other hand, Krasner's lemma asserts that for every such irreducible polynomial $P$, there exist a real number $c_P$ for every monic polynomial $Q$ such that the coefficients of $Q-P$ have absolute values $<c_P$, then $Q$ has a root in the field $E_P=\mathbf Q_\ell[T]/(P)$. By compactness of $\mathbf Z_\ell$, the set of all finite subextensions of given degree of $\overline{\mathbf Q_\ell}$ is finite. QED.

Let us now give the proof of the theorem. Let $(E_n)$ be a increasing sequence of finite subextensions of $\overline{\mathbf Q_\ell}$ such that $\overline{\mathbf Q_\ell}=\bigcup_n E_n$ (lemma 2; take for $E_n$ the subfield generated by $E_{n-1}$ and all the subextensions of degree $n$ of $\overline{\mathbf Q_\ell}$). Then $G_n=G\cap \mathop{\rm GL}_d(E_n)$ is a closed subgroup of $G$, and $G$ is the increasing union of all $G_n$. By lemma 1, there exists an integer $n$ such that $G_n=G$. QED.