Let $L/ \Bbb{Q}_p$ be finite extension, $o$ be it's ring of integers. Frobenius power series is defined as $Φ(X)∈o[[X]]$ s.t.$Φ(X)＝πX＋$(higher term) and $Φ(X)≡X^qmodπo[X]$.
It is well known that For given Frobenius power series,there exists unique formal group which has $Φ(X)$ as homomorphism.
For example, $Φ(X)＝(1＋X)^p-1$ is homomorphism of formal group $F(X,Y)＝X＋Y＋XY$.
My question is, does all formal group have Frobenius power series as homomorphism? For example, formal group $G(X,Y)＝X＋Y$ has Frobenius power series as homomorphism?
No, formal groups with endomorphisms of that shape are exceptional. These formal groups have in some sense a maximal ring of endomorphisms, while a typical formal group of finite height has only $\Bbb Z_p$ as endomorphism ring.
The infinite-height formal groups like the additive formal group $X+Y$ show just the opposite behavior: the endomorphisms defined over a ring $\mathfrak o$ of the sort you mention consist of all $\mathfrak o$: the endomorphism ring increases when $\mathfrak o$ is replaced by a larger ring, whereas the endomorphism ring of a finite-height formal group has $\Bbb Z_p$-rank bounded by the height.