# 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?

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.

• Perhaps your definition of “dimension” is different to mine, but by my definition, the dimension of the $p$-divisible of a $g$-dimensional Abelian variety is always just $g$. – Lubin Jun 18 '18 at 13:31
• Also I believe the formality in "formal $p$-divisible group" is in reference to formal groups -- look up formal schemes (formal groups are essentially formal schemes with additional structure) if you want an introduction to formal groups from a geometric point of view. – Jeff Yelton Jun 18 '18 at 15:12
• Thanks @Lubin and Jeff. I apologize for not being clear in my original question. I will 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" – aytio Jun 18 '18 at 22:37
• About the comment "higher height 1-dimensional groups, which shouldn't arise from abelian varieties": I think they do! It seems to me that they can arise as summands in the p-div'l group of an abelian variety, just not as the whole p-div'l group. For example, an abelian 3-fold over the algebraic closure of F_p can have p-div'l group G_{1/3} + G_{2/3}, where G_{r/s} has dimension r and height s. So there's a height 3 1-dim'l formal group showing up in the p-div'l group of an abelian variety. I really like chapter 4 of Demazure's "Lectures on p-div'l groups" as a reference for this material. – user124192 Jun 18 '18 at 23:33
• Perfectly right, @aaaaaaaaaaaaaaa. – Lubin Jun 19 '18 at 1:43

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.
• Thanks @Jeff. Unfortunately, that's not what I mean by dimension. I've edited the question to include the definition I am using. It is not merely the height of the connected part. For example over an algebraically closed field (perfect probably suffices) $K$, for every integer $h\ge1$, there is a unique connected 1-dimensional $p$-divisible group of height $h$, typically denoted $\Sigma_{K,h}$ (for example, page 3, Prop. 1 of its.caltech.edu/~mantovan/papers/Igusa.pdf ) – aytio Jun 18 '18 at 22:53