Let $p$ be a fixed prime.

Let $A$ be an abelian variety of $\mathbb{Q}_p$ and let $\mathcal{A}$ denote the Neron model of $A$ over $\mathbb{Z}_p$. Now, the multiplication by $p$ map in $\text{End}(A)$ extends to an endomorphism of $\mathcal{A}$, and let $\mathcal{A}[p]$ denote the (scheme-theoretic) kernel of this morphism.

Here are my questions on the properties of $\mathcal{A}[p]$:

  1. Is it always quasi-finite?
  2. Is it always flat?
  3. Is it always separated?

If $\mathcal{A}$ is a semi-abelian scheme, then $\mathcal{A}[p]$ is a quasi-finite flat separated group scheme by SGA 7, Exp IX, Lemme 2.2.1. I want to know what happens when $\mathcal{A}$ is not semi-abelian. For instance, what can we say about $\mathcal{E}[p]$ if $E$ is an elliptic curve over $\mathbb{Q}_p$ which has additive reduction?

  • 6
    $\begingroup$ If $\mathcal{E}\times_{\text{Spec}\ \mathbb{Z}_p}\text{Spec}\ \mathbb{F}_p$ is isomorphic to $\mathbb{G}_{a,\mathbb{F}_p} = \text{Spec}\ \mathbb{F}_p[t]$, then the kernel of the $p^{\text{th}}$ iterate of the group operation ("multiplication by $p$") is the entire closed fiber $\mathcal{E}\times_{\text{Spec}\ \mathbb{Z}_p}\text{Spec}\ \mathbb{F}_p$. Thus, the $p$-torsion group scheme $\mathcal{E}[p]$ over $\text{Spec}\ \mathbb{Z}_p$ is not quasi-finite, and it is not flat. As a closed subgroup scheme of the separated group scheme $\mathcal{A}$, always $\mathcal{A}[p]$ is separated. $\endgroup$ – Jason Starr Mar 14 '18 at 9:19

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