Exercise 9, § 2

With the notation of exerc. 7) one denotes by $\mathbf{O}_2^+(\mathbf{Q}_p)$ (the group of rotations of the space $\mathbf{Q}_p^2$) the group of matrices of the form

with elements in $\mathbf{Q}_p$, such that $x^2 + y^2 = 1$, this group being endowed with the topology defined in Gen. Top., VIII, p. 125, exerc. 2.

a) Denote by $G_n$ the subgroup of $\mathbf{O}_2^+(\mathbf{Q}_p)$ formed by the matrices such that $t = y/(1 + x) \in \mathfrak{p}^n$. Show that $G_n$ is a compact group, that $G_n / G_{n+1}$ is isomorphic to $\mathbf{Z}/p\mathbf{Z}$, and that the only compact subgroups of $G_1$ are the groups $G_n$ (cf. Gen. Top., III, p. 323, exerc. 24).

b) Show that $G_1$ is identical to the subgroup of matrices

such that $x^2 + y^2 = 1,\ x \in 1 + \mathfrak{p}^2$ and $y \in \mathfrak{p}$ if $p \neq 2$; to $x \in 1 + \mathfrak{p}^3$ and $y \in \mathfrak{p}^2$ if $p = 2$.

c) Show that the series with general terms $(-1)^n x^{2n}/(2n)!$ and $(-1)^n {}^1 x^{2n+1}/(2n+1)!$ are convergent for every $x \in \mathfrak{p}$ if $\neq 2$, for every $x \in \mathfrak{p}^2$ if $p = 2$; let $\cos x$ and $\sin x$ be the sums of these series. Show that the map

is an isomorphism of the additive topological group $\mathfrak{p}$ (resp. $\mathfrak{p}^2$) onto the group $G_1$.

d) If $p$ is of the form $4h + 1$ ($h$ an integer), there exists in $\mathbf{Q}_p$ an element $i$ such that $i^2 = -1$. If with every $z \in \mathbf{Q}_p^*$ one associates the matrix

one defines an isomorphism of the multiplicative group $\mathbf{Q}_p^*$ onto the group $\mathbf{O}_2^+(\mathbf{Q}_p)$; under this isomorphism the group $1 + \mathfrak{p}$ corresponds to $G_1$ and one has $\cos px + i \sin px = e_p^{t'}$ (III, p. 126, exerc. 8).

e) If $p$ is of the form $4h + 3$ ($h$ an integer) the elements of the matrices of $\mathbf{O}_2^+(\mathbf{Q}_p)$ are necessarily $p$-adic integers. The polynomial $X^2 + 1$ is then irreducible in $\mathbf{Q}_p$; let $\mathbf{Q}_p(i)$ be the quadratic extension of $\mathbf{Q}_p$ obtained by adjoining a root $i$ of $X^2 + 1$; one endows $\mathbf{Q}_p(i)$ with the topology defined in Gen. Top., VIII, p. 127, exerc. 2. The group $\mathbf{O}_2^+(\mathbf{Q}_p)$ is isomorphic to the multiplicative group $\mathbf{N}$ of elements of $\mathbf{Q}_p(i)$ of norm 1, under the isomorphism which associates the matrix $\begin{pmatrix} x & y \\ -y & x \end{pmatrix}$ with the element $z = x + iy$. Show that there are $p + 1$ roots of the equation $x^{p+1} = 1$ in $\mathbf{Q}_p(i)$ and that they form a cyclic subgroup $\mathbf{R}$ of $\mathbf{N}$ (argue as in Gen. Top., III, p. 323, exerc. 24; then show that there are $p + 1$ distinct roots of the congruence $x^{p+1} \equiv 1$ (mod. $p$) in $\mathbf{Q}_p(i)$, and, for each root $a$ of this congruence, form the series $(a^{p^{2n}})$). Deduce that the group $\mathbf{O}_2^+(\mathbf{Q}_p)$ is isomorphic to the product of the groups $\mathbf{R}$ and $\mathbf{G}_1$.

f) Show that, for $p = 2$, the group $\mathbf{O}_2^+(\mathbf{Q}_p)$ is isomorphic to the product of the group $\mathbf{G}_1$ and the cyclic group of order 4.