Let $\Gamma(p) := \text{ker}(SL\_2(\mathbb{Z}\_p)\rightarrow SL\_2(\mathbb{Z}\_p/p))$.


Viewing $SL\_2(\mathbb{Z}\_p)$ as an analytic group, is there a formal group law $F$ in three variables, defined over $\mathbb{Z}\_p$, such that $\Gamma(p)$ is isomorphic to the group on the set $(p\mathbb{Z}\_p)^3$ whose group operation is given by $F$?


Here, "standard" is the terminology used in Serre's lecture notes on Lie Algebras and Lie Groups.



Is $\Gamma(p) := \text{Ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{F}_p)$ a "standard" subgroup?
Yes. Let $g = \begin{pmatrix}1+a & b\\ c & 1+d\end{pmatrix} \in \Gamma(p)$ with $a,b,c,d\in p\mathbb{Z}\_p$.
Then the condition $\det g = 1$ can be rewritten $(1+a)(1+d) = bc+1$, so that $d = (1+a)^{-1}(bc+1) -1$, which is a power series in $a,b,c$ with zero constant term. Then you can turn the group law and the inverse expressed as polynomials in $a,b,c,d$ into power series in $a,b,c$.


