3
$\begingroup$

Let $p$ be a prime. Let $K/\mathbb{Q}_p$ be a finite extension and $\mathcal{O}=\mathcal{O}_K$ its ring of integers.

Let $A_K$ be an abelian variety and $\mathcal{A}_\mathcal{O}$ denote its Neron model over $\mathcal{O}$. We don't assume that $\mathcal{A}_\mathcal{O}$ has good or semistable reduction.

Let $\ell$ be a prime different from $p$ and let $G=A[\ell]_\mathcal{O}$ be the scheme-theoretic kernel of multiplication by $\ell$ in $\mathcal{A}_\mathcal{O}$. We know that $G\times_{\text{Spec} ~\mathcal{O}} \text{Spec} ~K = A_K[\ell]$ is a finite subgroup scheme of order $\ell^{2g}$, where $g$ is the dimension of $A_K$.

I have a very stupid question here:

Is $A[\ell]_{\mathcal{O}}$ finite and flat? (or at least quasi-finite)? If not, are there some necessary conditions for $A[\ell]_{\mathcal{O}}$ to be finite and flat?

$\endgroup$
  • $\begingroup$ See 20.7 of Milne, J. S. Abelian varieties. Arithmetic geometry (Storrs, Conn., 1984), 103--150, Springer, New York, 1986. MR0861974, if $A_\mathcal{O}$ has good reduction. $\endgroup$ – user1225 Nov 25 '17 at 3:18
  • $\begingroup$ If the generic fiber $A_K$ is an elliptic curve and the closed fiber is a multiplicative group, then the generic fiber of $A[\ell]_{\mathcal{O}}$ has length $\ell^2$, yet the closed fiber has length $\ell$. Thus, the affine, flat group scheme $A[\ell]_{\mathcal{O}}$ over $\text{Spec}\ \mathcal{O}$ is not finite (it is quasi-finite). $\endgroup$ – Jason Starr Nov 25 '17 at 3:34
  • $\begingroup$ See Lemma 2 of Section 7.3 of S. Bosch, W. Lutkebohmert, M. Raynaud, Neron Models (1990). If $A$ has semistable reduction, then $A[\ell]_{\mathcal{O}}$ is quasi-finite and flat. $\endgroup$ – user1225 Nov 25 '17 at 3:41
8
$\begingroup$

In general the scheme $\mathcal A[l]_{\mathcal O}$ is not finite because of the following lemma.

Let $f:X\to Y$ be a separated quasi-finite flat morphism of noetherian schemes. Then it is finite iff the fibral rank is locally constant.

For a proof see the paper "Les schémas de modules des courbes elliptiques" by Deligne and Rappoport (II$.1.19$).

So, you actually need just to compute the fibral rank of $\mathcal A[l]_{\mathcal O}$. For example, if $A_K$ is an elliptic curve with additive or multiplicative reduction it is usually not finite. Note that there is a small mistake in Jason Starr's comment, the rank of $\mathcal A[l]_k$ might be bigger than $l$ in the case of multiplicative reduction b/c $\mathcal A[l]_k$ isn't necessary connected.

However, $\mathcal A[l]_{\mathcal O}$ is always quasi-finite provided that $l$ is invertible on $\mathcal O_K$. Moreover, it is always étale and you don't need any semi-stability assumptions for this claim. For a proof look at Lemma 7.3/2 in the book "Néron Models" by Bosch, Lutkebohmert, Raynaud.

$\endgroup$
  • $\begingroup$ For the result that is highlighted, either assume $Y$ is connected or replace "constant" with "locally constant". $\endgroup$ – nfdc23 Nov 25 '17 at 4:20
  • $\begingroup$ (also, maybe change "sch´emas" to "schémas") $\endgroup$ – Noam D. Elkies Nov 25 '17 at 4:55
  • $\begingroup$ Oops, I forgot about the component group of the closed fiber. $\endgroup$ – Jason Starr Nov 25 '17 at 5:25
  • $\begingroup$ Thank you very much! For instance, if we take $\mathcal{A}=X_0(11)$, I think $\mathcal{A}[5]$ is finite even though $\mathcal{A}$ has multiplicative reduction at $11$, which is an example. $\endgroup$ – user1225 Nov 25 '17 at 5:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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