Let $G$ be an affine algebraic group and $X$ be an abelian variety over $Spec(R)$, where $R$ is a commutative noetherian ring. Let $\widehat{G}$ and $\widehat{X}$ be the completion along the identity section $\epsilon_G : Spec(R) \rightarrow G$ and $\epsilon_X : Spec(R) \rightarrow X$ respectively. Then one can see that $\widehat{G}$ and $\widehat{X}$ are formal groups. It is clear that a homomorphism of groups give a homomorphism of the associated formal groups. Since $G$ is affine and $X$ is an abelian vairety there cannot be any non-trivial morphism from $G$ to $X$. My question is the following :
Question 1. Does there exist groups $G$ and $X$ and a non-trivial morphism (an isomorphism) $\phi : \widehat{G} \rightarrow \widehat{X}$? In particular, I would like to know if one can detect whether the group is affine or projective using the associated formal groups.
Question 2. In the paper titled $p$-divisible groups, the author J. Tate raised the following question : Are there any $p$-divisible groups over $\mathbb{Z}$ other than the products of powers of $\mathbb{G}_m(p)$ and of $\mathbb{Q}_p/\mathbb{Z}_p$? I suppose this has been asked in light of the fact that there are no abelian schemes over $\mathbb{Z}$ and then only one-dimensional affine algebraic groups are $\mathbb{G}_a$ and $\mathbb{G}_m$. The $p$-divisble group assiated to $\mathbb{G}_a$ is Is that correct? What is currently known about this problem?
The two questions seem rather disconnected but I suppose they can be put down under the same ubmrella of $p$-divisible groups and hence the reason for putting them down as one question.
Thank you for your time.
This is a (partial?) answer to your Question 1 only.
Since you have left your ring $R$ completely unspecified, I’ll give an example when $R$ is an algebraically closed field of characteristic $p>0$. Perhaps you knew this special case already?
The example takes $G$ as the multiplicative group $\mathbf G_{\text m}$ and $X$ as an ordinary elliptic curve over the field $R$. Then their formal groups, being both of height one, are $R$-isomorphic.