1-dimensional p-divisible groups, level structures and Cartier divisors I am confused about the 1-dimensionality of $p$-divisible groups and its role in defining level structures.
Here's how I view/understand/not understand things:
If a $p$-divisible group arises from a dimension $g$ abelian variety (say over some $S$ over $\mathbb F_p$), then it is of height $2g$ and dimension at least $g$, with equality in the ordinary case.
So for $g>1$, such $p$-divisible groups are never 1-dimensional and if $g=1$, they are of height 2.
On the other hand, from what I've been reading, whenever a good notion of level structure is mentioned, the assumption is usually that the $p$-divisible group is 1-dimensional. I am still confused as to why. I understand it should be related to the fact that Cartier divisors make sense but am not entirely sure what is the ambient curve since such $p$-divisible groups do not arise from abelian varieties.. 
I am familiar with Katz-Mazur's definition of level structure/full set of sections, I understand Drinfeld modules and the notion of level structure (in that case the Drinfeld module is 1-dimensional over the base so Cartier divisors make sense etc). I am however confused how this all relates among each other... in the etale case a level structure seems to be a choice of isomorphism with the constant group scheme, but then there is also a notion of level structure for formal $p$-divisible groups, and I've usually interpreted (maybe erroneously?) "formal" roughly as being "connected"?; but then over a perfect base (say a perfect field) there are no sections and then any level structure is trivial?.. I really hope this brief rambling exposes to an expert where my confusion is..
As an example of an explicit question, on page 20 of https://arxiv.org/pdf/1005.2558.pdf to a Drinfeld level structure $\varphi:\mathbb F_p^d\to X_0[p]$ a filtration is defined by the equality of divisors $[H_i]=\displaystyle\sum_{x\in\text{span}(e_1,\ldots,e_i)}[\varphi(x)]$. Q: Where do these divisors "live" and where exactly is 1-dimensionality used?
Any illuminating comments/answers would be greatly appreciated.
Thank you.
Edit: By dimension of a $p$-divisible groups I mean the (locally constant) rank of the Lie algebra (such as in Messing, or page 59 of Harris-Taylor "The Geometry and Cohomology of Some Simple Shimura Varieties"). In particular the height can be any integer $h\ge1$. The arxiv paper referenced above works with higher height 1-dimensional groups, which shouldn't arise from abelian varieties, yet speaks of Cartier divisors, hence my confusion as I don't know on which ambient scheme these live on.
 A: This is not a full answer -- in particular, I'm not familiar enough with Drinfeld modules to comment on that part of your question -- but perhaps I can clear up a bit of your confusion.
It's true that the height of the $p$-divisible group $G$ coming from an abelian variety of dimension $g$ is always equal to $2g$, while the height $h$ of the identity component $G_0$ of the $G$ satisfies $g \leq h \leq 2g$ (the extremes $h = g$ and $h = 2g$ correspond to the ordinary and supersingular cases respectively). This reflects the fact that the kernel of the multiplication-by-$p$ map has order $p^{2g}$, but pointwise (the etale part) it has some lower $\mathbb{Z} / p\mathbb{Z}$-rank, namely $2g - h$. It seems that what you are defining the dimension is what sources I've seen define as the height of $G_0$, and the dimension is actually always $g$ as Lubin has commented.
Then the reason sources that discuss level structures tend to assume dimension $1$ is probably that most study of moduli spaces of level structures treats the elliptic curve (or $1$-dimensional) case.
