Let $R$ be a commutative ring, and, for $n\ge0$,
${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series
$u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which
$a_0\in R^\times$ and $u(x)\equiv x\pmod{x^{n+1}}$.
The group operation is composition of series.
A series in ${\mathcal{A}}_n$ that’s not in ${\mathcal{A}}_{n+1}$
is said to have depth $n$. When $\kappa$ is a field with $p^s$ elements,
${\mathcal{A}}_1(\kappa)$ is often called the Nottingham
group, but it seems to have been designated by different
names in different times and places.
One sees that ${\mathcal{A}}_n$ is the projective limit of
the groups ${\mathcal{A}}_n/{\mathcal{A}}_{n+r}$, and for $n>0$
these are unipotent algebraic groups of dimension $r$. So
it's the proalgebraic group ${\mathcal{A}}_1$, defined over
${\mathbb{F}}_p$, that I'm asking
about, and the existence of (finite-dimensional) algebraic
subgroups in it.
There's one that's obvious, namely the image of
${\mathbf{G}}_{\mathrm a}\to {\mathcal{A}}_1$ by
the homomorphism
$$
t\mapsto \frac{x}{1-tx}
=x+tx^2+t^2x^3+\cdots=u_t(x)
$$
fractional-linear transformation leaving the origin fixed,
expanded as series.
Almost as obvious is to apply the transformation $$ u(x)=x(1+g(x))\mapsto (u(x^m))^{1/m} =x(1+g(x^m))^{1/m} $$ clearly a group homomorphism, well defined when $m$ is prime to $p$, because you're raising a principal unit to an exponent that's in ${\mathbb{Z}}_p$. This takes $u_t$, a subgroup of Nottingham of depth one to a group of depth $m$. In particular, the Nottingham proalgebraic group has lots of one-dimensional algebraic subgroups.
There are no commutative connected algebraic subgroups of dimension greater than $1$: I've found an argument that depends on higher ramification theory and uses Hasse-Arf, but breaks down completely in any noncommutative case. I would like to settle the question of whether there are any algebraic subgroups of Nottingham of dimension greater than $1$, necessarily noncommutative: it will be enough to dispose of the two-dimensional case. So it occurs to me that this may already be known, or maybe someone out there can suggest an approach to me.
