# Moduli Space of Abelian Varieties with a N-torsion point

Does there exists (as scheme, or as some sort of stack) the moduli space of principally polarized Abelian Varieties together with a point of order $N$, for $N>1$ an integer?

In the case of dimension 1, it is the well known modular curve $X_1(N)$; but for abelian varieties I have seen essentially only the corresponding to the full modular curve $X(N)$ in dimension 1. Are there some obstructions (bad behaved schemes, bad compactifications, or any other reasons) for considering these moduli spaces?

• Note that this always fails to be a scheme because some points have extra automorphisms: take an abelian variety one dimension less with a point of order $N$ with a curve with extra automorphisms. – Will Sawin Jun 28 '12 at 2:34

No, there are no problems. The stack of principally polarized abelian varieties $\mathcal A_g$ has a universal family $\mathcal X_g \to \mathcal A_g$, which is a relative group scheme. As such, it has an endomorphism $\mathcal X_g \to \mathcal X_g$ representing multiplication by $N$, which is finite and flat. Its kernel is the stack you want; the map to $\mathcal A_g$ is finite and flat. In characteristic prime to $N$, it is also étale.
 I was too hasty. As Kevin points out, one should take points of order $N$ in the kernel. This is an open and closed substack when $N$ is invertible. Over $\mathbb Z$, the kernel will have several irreducible components that will meet over primes dividing $N$, and it is not so clear to me how to distinguish them. For example, suppose that $N = p$, where $p$ is a prime; then all the components of the kernel will intersect over the locus of abelian varieties with $p$-rank $0$. It is not clear to me how to give a modular interpretation of the closure of the locus of points of order $N$ over $\mathbb Z[1/N]$.
• As I remember, the question of defining "points of exact order $N$" is addressed when $g=1$ by Katz and Mazur, "Arithmetic moduli of elliptic curves". If $g>1$, I don't know if there is any reasonable answer. The nice thing when $g=1$ is that you can view sections as relative divisors. – Laurent Moret-Bailly Jun 28 '12 at 20:11
To supplement Angelo's answer, you should be able construct it as analytic space by taking the quotient of the Siegel upper half plane $H_g$ by the subgroup of matrices $M\in Sp_{2g}(\mathbb{Z})$ with first column congruent to $e_1=(1,0,\ldots 0)^T\mod N$. Given $\Omega\in H_g$, its image corresponds to the abelian variety with $N$-torsion point $(\mathbb{C}^g/\mathbb{Z}^g+\Omega\mathbb{Z}^g, \frac{1}{N}e_1)$.