Exercise 6, § 7

In this exercise, $l$ denotes a prime number $\neq 2$. If $a$ is an integer $\neq 0$, let $v_l(a)$ denote the $l$-adic valuation of $a$, i.e. the largest integer $e$ such that $a \equiv 0 \pmod{l^e}$.

(a) Let $m$ be an integer $\geq 1$. We write

Show that, if $x$ is an integer prime to $l$, then and that there is equality if the image of $x$ in the cyclic group $(\mathbf{Z}/l^2\mathbf{Z})^*$ is a generator of this group.

(b) Let $n$ be an integer $\geq 1$. We write

Show that

where the symbol $[\alpha]$ denotes the integral part of the real number $\alpha$. Show that, if $x$ is an integer prime to $l$, then

and that there is equality if the image of $x$ in $(\mathbf{Z}/l^2\mathbf{Z})^*$ is a generator of this group.

(c) In the notation of Exercise 5 (c), show that $l^{r(l, n)}$ is the greatest power of $l$ which divides all the $a_n(p)$, for $p$ prime $\neq l$ (or for all sufficiently large $p$, which amounts to the same). (Use (b) applied to $x = p$; then choose $p$ such that its image in $(\mathbf{Z}/l^2\mathbf{Z})^*$ is a generator of this group, which is possible by the arithmetic progression theorem.\footnote{For a proof of this theorem, see for example A. Selberg, Ann. of Math., 50 (1949), pp. 297–304.})

(d) Let $\Gamma$ be a finite subgroup of $\mathbf{GL}(n, \mathbf{Q})$ and let $l^e$ be the greatest power of $l$ dividing the order of $\Gamma$. Show that $e \leq r(l, n)$. (Use Exercise 5 to prove that $l^e$ divides all the $a_n(p)$ and then apply (c) above.)

(e) Conversely, show that there exists a finite $l$-subgroup $\Gamma_{l, n}$ of $\mathbf{GL}(n, \mathbf{Q})$ whose order is $l^{r(l, n)}$. (Reduce it to the case where $n$ is of the form $l^a(l-1)$, with $a \geq 0$. Decompose $\mathbf{Q}^n$ as a sum of $l^a$ copies of $\mathbf{Q}^{l-1}$ and use this decomposition to give an operation on $\mathbf{Q}^n$ of the semi-direct product $H_{l, n}$ of the symmetric group $S_{la}$ and the group $(\mathbf{Z}/l\mathbf{Z})^{la}$. Take $\Gamma_{l, n}$ to be a Sylow $l$-group of $H_{l, n}$.)

Show that, if $n$ is even, $\Gamma_{l, n}$ is contained in a conjugate of the symplectic group $\mathbf{Sp}(n, \mathbf{Z})$.

(f) Let $\Gamma$ be a finite subgroup of $\mathbf{GL}(n, \mathbf{Q})$ which is an $l$-group. Show that $\Gamma$ is conjugate to a subgroup of $\Gamma_{l, n}$. (Show first, using a suitable reduction modulo $p$, that the reduction of $\Gamma$ is a finite subgroup of the reduction of a conjugate of $\Gamma_{l, n}$; then use the character of the representation of $\Gamma$ on $\mathbf{Q}^n$.) In particular, every subgroup of $\mathbf{GL}(n, \mathbf{Q})$ of order $l^{r(l, n)}$ is conjugate to $\Gamma_{l, n}$