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Let $A$ be an abelian variety defined over a number field $K$ with good reduction everywhere. Let $\mathcal P$ be a prime of $K$.

Does reduction of $A$ modulo $\mathcal P$ induce a well-defined reduction map from the tangent space of $A$ at identity to the tangent space at identity of the reduced variety?

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The only natural thing I can think of is the following:

Let $R_{\mathcal{P}}$ be the ring of integers of $K_{\mathcal{P}}$ (the completion at the prime). Let $S$ be the spectrum of $R_{\mathcal{P}}$. Since $A$ has good reduction, there is an abelian scheme (the Néron model) $\mathcal{A}$ over $S$, with $A$ as generic fibre, and the reduction as special fibre (which we will denote $A_{0}$.

Now look at maps $S[\varepsilon] \to \mathcal{A}$ that restrict to the identity section $e \colon S \to \mathcal{A}$, where $S[\varepsilon]$ is $S \times \mathrm{Spec}(\mathbb{Z}[\varepsilon]/(\varepsilon^{2}))$. Such maps $S[\varepsilon] \to \mathcal{A}$ restrict to tangent vectors on the generic fibre and the special fibre. So you get maps $$ T_{e}A \longleftarrow \mathrm{Hom}_{S}(S[\varepsilon],\mathcal{A}) \longrightarrow T_{e}A_{0}. $$

Like René observed below in the comments:

the middle term is a tangent space in its own right (and a free $R_{\mathcal{P}}$-module of rank $\dim A$), which naturally embeds into $T_{e}A$ and has a reduction map to $T_{e}A_{0}$.

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  • $\begingroup$ Could you expand a bit more on the naturality? There seems to be a choice of abelian scheme. $\endgroup$
    – Matthias Wendt
    May 7, 2015 at 9:47
  • $\begingroup$ The definition of "good reduction" is that there is such an abelian scheme. And won't $\mathcal A$ be unique (as an abelian scheme over $R_{\mathcal P}$)? Actually, even if $A$ has bad reduction, one can specify that $\mathcal A$ be the Neron model, which will pin things down. $\endgroup$
    – Joe Silverman
    May 7, 2015 at 10:50
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    $\begingroup$ With $\mathrm{Hom}_{S}(S[\varepsilon],\mathcal{A})$, do you mean the set of those $S$-morphisms $S[\varepsilon]\to\mathcal{A}$ that restrict to the identity section $e\colon S\to\mathcal{A}$? And in that case, wouldn't the middle term just be "the tangent space to $\mathcal{A}$ at the identity section" (i.e. something like $\operatorname{H}^0(S,e^\ast (\Omega^1_{\mathcal{A}})^{\vee})$)? $\endgroup$
    – R.P.
    May 7, 2015 at 11:16
  • $\begingroup$ @René — You are right, I forgot to specify that. $\endgroup$
    – jmc
    May 7, 2015 at 11:52
  • $\begingroup$ @MatthiasWendt,JoeSilverman — Yes, I should have mentioned the Néron model. $\endgroup$
    – jmc
    May 7, 2015 at 11:52
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There is no such natural map. Note that both the tangent spaces you are talking about are vector spaces over different fields.

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  • $\begingroup$ I know that they are vector spaces over different fields. I was wondering if a "good choice" of local parameters at identity of $A$ reduces to a set of local parameters at the identity of the reduced variety. $\endgroup$
    – Shaki
    May 7, 2015 at 8:52

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