I guess what might be still confusing, is that F may not immediately be recognized as a functor of points, that is it has a description which doesn't require G. There likely is a better way to ask this, but hopefully its roughly clear what I'm concerned with.

Hi! Does any body has the zoom link and password for motives and what not?

Is there any place where I can find a description of the Cartier duals of a height $n > 0$ formal group over a perfect field k? (For example, things like the dimension, which I believe is $n-1$.)

Actually, the dimensions of a p-divisible group and its Cartier dual necessarily add up to the height, so that solves this. Let me make a more precise question.

Suppose $\mathbb{G}$ is a formal finite height group, then $\mathbb{G}[p]$ is a finite group scheme of rank $p^{n}$. What is the coordinate ring of its Cartier dual?

(Note that it is necessarily a tensor product of an etale algebra and local algebras of the form $k[x]/(x^{q})$, where $q$ is some power of $p$. I would like to know these exponents.)

For example, if $\mathbb{G} = \mathbb{G}_{m}^{\vee}$ is the formal multiplicative group, then the Cartier dual of $\mathbb{G}[p]$ is the constant group scheme $\mathbb{Z}/p$, with coordinate ring $map(\mathbb{Z}/p, k)$.

@PiotrPstrągowski if you take the hopf algebra of functions on the original finite free group scheme, take its linear dual, and forget down to the resulting ring, you get the coordinate ring on the Cartier dual. for the height n Honda formal group, you can find the answer in section 5 of Ravenel-Wilson's ... Conner-Floyd Conjecture: it's spanned by elements 1, beta_(0), beta_(1), ..., beta_(n) subject only to the relations beta_(j)^p = beta_(j-n+1) and beta_{negative} = 0

this is a pretty good approximation to the formula for any ol' height n formal group