# Local Norm Mapping for Abelian Varieties

Let $A/K$ be an abelian variety defined over a nonarchimedean local field $K$ of characteristic $0$ and let $L$ be a finite extension of $K$. Consider the norm map $$A(L)\xrightarrow{N_{L/K}}A(K)$$ I want to know if the following statement is true -

If $A_{0}(K)$ is the subgroup of $A(K)$ specializing to the connected component of the Neron model, then $$\bigcap_{L}N_{L/K}A(L)\cong A_{0}(K)$$ where the intersection is over all finite unramified extensions $L$.

When $A$ has non-degenerate reduction over $K$ (i.e. the dual variety ${A}^{\vee}$ has good reduction), this follows immediately from Tate Local Duality and Corollary 4.4 in Mazur's 1972 paper 'Rational Points of Abelian Varieties with Values in Towers of Number Fields.' Does it also hold when $A$ does not have non-degenerate reduction over $K$?

One doesn't need Tate local duality to analyze the good reduction case, and in general the answer is affirmative.

First, let's review the general nonsense for norm maps with commutative group functors (to sidestep representability issues). For any finite etale map of schemes $f:S' \rightarrow S$ and any commutative functor $F$ on the category of $S$-schemes there is a natural "norm" map of group functors $N: f_{\ast}(F_{S'}) \rightarrow F$, where $F_{S'}$ denotes the restriction of $F$ to the category of $S$-schemes (represented by base change when $F$ is representable). Indeed, if $T$ is an $S$-scheme then $f_{\ast}(F_{S'})(T) = F(T \times_S S')$ is identified with the group of global sections of the $f_T$-pullback of the restriction of $F_T$ to the small etale site over $T$, so to define $N$ on $T$-points we simply use the natural trace map $(f_T)_{\ast} \circ f_T^{\ast} \rightarrow {\rm{id}}$ on the category of abelian sheaves on the small etale site of $T$. (This trace map is defined more generally using just that $f$ is quasi-finite flat and separated and finitely presented, but later we will need that $f$ is finite etale and the definition of the trace map is much simpler in the case of finite etale $f$ anyway.)

For example, taking $F$ to be the functor of points of the Neron model $\mathcal{A}$ of $A$ over $O_K$ and $f$ to correspond to $O_K \rightarrow O_L$, the above norm map on $O_K$-points is precisely the natural norm map $A(L) \rightarrow A(K)$ due to the universal mapping property of Neron models and the compatibility of Neron models with unramified base change. The advantage of this geometric interpretation is that the norm map of smooth finite type $O_K$-groups $$N_{A,L/K}:{\rm{R}}_{O_L/O_K}(\mathcal{A}_{O_L}) \rightarrow \mathcal{A}$$ is an $O_K$-form of the addition map $\mathcal{A}^{[L:K]} \rightarrow \mathcal{A}$. More generally, for any $f$ and $F$ as in the preceding paragraph with $f$ of constant degree $d$, the "norm" map $f_{\ast}(F_{S'}) \rightarrow F$ becomes the addition map $F^{\oplus d} \rightarrow F$ after passing to an etale cover of $S$ that splits $S'$ into a disjoint union of $d$ copies of $S$ (a fiber power of $S'$ over $S$ is such a cover). Indeed, the formation of the norm commutes with base change on $S$, and in the case of a split covering it is clear from the construction that the norm is identified with the componentwise addition map.

Now we see that by taking $[L:K]$ divisible by the order of the finite etale component group of the special fiber $\mathcal{A}_0$, $N_{A,L/K}$ lands inside the relative identity component $G := \mathcal{A}^0$ (open complement of the closed union of the finitely many non-identity components in the special fiber) since such an assertion is local for the etale topology on $O_K$ and hence it can be checked by computing with the $O_K$-form of the map given by componentwise addition $\mathcal{A}^{\oplus [L:K]} \rightarrow \mathcal{A}$. Hence, we may replace $\mathcal{A}$ with $G$ and reduce to checking that the norm map $$N_{G,O_L/O_K}:{\rm{R}}_{O_L/O_K}(G_{O_L}) \rightarrow G$$ is surjective on $O_K$-points for any smooth $O_K$-group $G$ of finite type with connected fibers.

The kernel $G' = \ker N_{G,O_L/O_K}$ is a form of $G^{[L:K]-1}$ for the etale topology on $O_K$, so it is also smooth of finite type with connected fibers. Hence, for any $O_K$-point $g$ of $G$, the $N_{G,O_L/O_K}$-pullback of $g:{\rm{Spec}}(O_K) \rightarrow G$ is a smooth $O_K$-scheme $X$ of finite type that is a $G'$-torsor for the etale topology, and we just need to show that $X(O_K)$ is non-empty. Since $X$ is smooth and $O_K$ is henselian (e.g., complete), such non-emptiness holds provided that the special fiber $X_0$ has a rational point over the residue field $k$ of $O_K$.

We have come down to the task of showing that any torsor for a smooth connected finite type group scheme over $k$ must have a $k$-point. So far there is no number theory in any of this, just general arithmetic geometry over discrete valuation rings. But finally we bring in finiteness of $k$ to apply Lang's theorem which asserts exactly that such torsors always have a rational point when the ground field $k$ is finite.

• "This trace map is defined more generally using just that f is quasi-finite flat and separated and finitely presented" : I'm very interested to see the details of the construction of the norm in such generality. – Matthieu Romagny Dec 3 '16 at 21:52
• Note: in the answer to this question, user27920 writes that "beyond the separable setting there is no "reasonable" norm through finite flat maps whose formation commutes with base change" which seems to go against your claim. – Matthieu Romagny Dec 3 '16 at 21:59