# Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.

Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]$ classifying abelian varieties with $N$-level structure.

Is there a finite proper map $A_g(N)\to A_g$ over $\mathbb Z$ with a nice moduli interpretation, which recovers the usual notion of $N$-level structure over $\mathbb Z[\frac 1N]$?

Remark: In fact, I believe that such an $A_g(N)\to A_g$ is determined canonically from $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]$ by "normalizing in the function field", so the content of the question is really to have a good moduli interpretation for this canonical extension.

What is the "right" notion of $p$-level structure on an abelian variety in characteristic $p$?

The impression I get from reading in various places is that the answer to this is well-known, but I haven't been able to find a good reference. I don't have a specific type of level structure in mind. I'd just like to know what is known (and where to find it) for some common notions of level structure. EDIT: Actually, for level structures consisting of a choice of subgroup(s), there is a natural choice of moduli space over $\mathbb Z$ classifying abelian varieties with a choice of subgroup scheme(s) (or equivalently, isogenies) and this is valid in "bad" characteristics -- see, e.g. the end of Chai. So, let me state my question for level structures given by choices of point(s), e.g. "a point of exact order $N$", or full level $N$ structure.

For elliptic curves, Katz--Mazur showed that the answer is provided by the notion of a Drinfeld level structure (first considered by Drinfeld). For example, the "right" notion of a $p$-torsion point on an elliptic curve $E\to S$ (for any scheme $S$) is a point $P:S\to E$ so that:

1. $pP=1$.
2. The Cartier divisor $\sum_{i=0}^{p-1}[iP]$ is a subgroup scheme.

As Katz--Mazur write, it is not clear how to generalize this notion to abelian varieties since points are no longer divisors.

• +1: very nice question! – Daniel Miller Dec 28 '13 at 20:09
• Note that $A_g$ classifies principally polarized abelian schemes (of relative dimension $g$); the polarization aspect comes along for free uniquely when $g=1$ but not otherwise, as you undoubtedly know. I was once told that there is a paper by Chai and Norman (title I do not know) which shows in some sense that there isn't a good version of Drinfeld's idea beyond relative dimension 1. Perhaps email your question to Chai (or Kai-Wen Lan) if you don't receive a satisfactory answer here. – user76758 Dec 28 '13 at 20:19
• The paper of Chai and Norman user76758 alludes to is likely this one: jstor.org/stable/2374734 – Daniel Litt Dec 28 '13 at 21:02
• Dear Daniel Litt: Thanks for tracking it down; that is indeed the right paper. It is also worth noting (in view of the question posed) that this paper provides a sense in which mere normalization in higher level (which doesn't have positive-dimensional fibers) is the "wrong" thing to consider, suggesting that perhaps the question posed about normalization is not a "useful" point of view for applications with $g > 1$. – user76758 Dec 28 '13 at 21:53