TITLE: Endomorphisms of the multiplicative formal group law
QUESTION [6 upvotes]: Is there a simple description of the ring of endomorphisms $\mathrm{End}(\mathbb{G}_m)$ of the formal group law $$\mathbb{G}_m(X,Y) = X + Y + XY,$$ at least over a ring of characteristic zero? I'm working with coefficients in $\mathbb{Z}_p$, $p$ a prime.
It contains at least the polynomials $(1+X)^n - 1$, $n \in \mathbb{N}$, but I'm not sure what else.

REPLY [5 votes]: I’m not sure of the right answer to a question of this generality. As mercio correctly notes, if you’re looking for $\Bbb Z_p$-endomorphisms, the answer is $\Bbb Z_p$ (we’re referring to the ring of $p$-adic integers here, of course). And if you’re looking for the $\mathscr O$-endomorphisms, where that’s the ring of integers in an algebraic extension of $\Bbb Q_p$, the answer is still $\Bbb Z_p$, as you can read in the text of Fröhlich. But if $R$ is any subring of $\mathbb Q$, it looks to me as if $\text{End}_R(\Bbb G_m)$ will again be $R$. What about if you take a subring $R$ of an algebraic number field $K$? I think the $R$-endomorphisms of the multiplicative formal group might turn out to be $R\cap\mathbb Q$. It might be a fun exercise to work this out.