3
$\begingroup$

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?

$\endgroup$
1
  • 1
    $\begingroup$ 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. $\endgroup$
    – Will Sawin
    Jun 28, 2012 at 2:34

2 Answers 2

10
$\begingroup$

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.

[Edit] 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 $\mathbb Z[1/N]$.

$\endgroup$
2
  • 1
    $\begingroup$ [It's not quite the kernel, it's the points of order N in the kernel, so some closed substack] $\endgroup$ Jun 27, 2012 at 19:55
  • 4
    $\begingroup$ 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. $\endgroup$ Jun 28, 2012 at 20:11
5
$\begingroup$

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 $(\mathbb{C}^g/\mathbb{Z}^g+\Omega\mathbb{Z}^g, \frac{1}{N}e_1)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.