In this section we study the interplay between N´eron functions associated to a non-archimedean placev and the N´eron model of A over the spectrum of the ring of integersOv of the completion kv of kat v. Our main references are [9] and [59].
Let R denote a Dedekind domain with field of fractions l and let S = Spec(R).
Definition 1.12. LetV be a smooth projective variety over lof dimension d. We call a closed subscheme in someP¯nl given by a set of defining equations ofV inPn¯l amodel ofV over Spec(l) and we say that the model isR-integral if the equations have coefficients inR. IfM is anR-integral model ofV over
Spec(l), then we call the closed subscheme of PnR defined by the equations inM theclosure of M over S.
Moreover, we define a model of V over S to be a normal and flat S-scheme V → S of dimension d+ 1 together with an isomorphism Vl ∼= V, where Vl is the generic fiber of V. For each closed v ∈ S we denote the special fiber of V above v byVv.
Remark 1.13. IfM is a model of a smooth projective varietyV over Spec(l), then we usually call M a model of V without mentioning Spec(l) explicitly.
Moreover, we will regularly abuse notation by usingV for both the variety and its given model, and talk about the closure ofV when we mean in fact the closure of the given (R-integral) model of V over S, unless this might cause confusion. Conversely, we always mention the base schemeSwhen we talk about a model of a variety V over S.
Note that if the closure of a given R-integral model of V is normal and flat, then it is a model ofV over S.
We are especially interested in models which are proper and regular.
However, ifV =Ais an abelian variety, then it is natural to look for models of A over S which are regular (or even smooth over S), but also retain as much of the group structure of Aas possible. It turns out that in general it is not possible to find such a model if we also require properness, but N´eron found a way to construct a model that satisfies a property which suffices in applications.
Definition 1.14. LetAbe an abelian variety defined overl. AN´eron model ofA over Sis a separated schemeA −→S with generic fiberAlisomorphic to Athat is smooth over S and satisfies the following universal property: If X −→ S is a smooth S-scheme with generic fiber Xl, then any morphism φ:Xl−→ Al extends uniquely to a morphismX −→ Aover S.
In particular, the uniqueness property guarantees that any l-rational point corresponds to a section inA(S). Although this is weaker than proper-ness, it suffices for most purposes. The next result states that N´eron models exist and that they have a structure which is as close to the group structure on Aas possible.
Theorem 1.15. (N´eron) Let A be an abelian variety defined over l. Then there exists a N´eron model A −→ S of A. It is a group scheme over S whose group scheme structure extends the Spec(l)-group scheme structure on A. Moreover it is unique up to unique isomorphism.
Proof. The original proof is very deep and can be found in [77]. For a more modern proof see [9].
We only use N´eron models locally, so we might as well restrict to the case whereR is a discrete valuation ring with field of fractionsl, valuation
vand residue fieldl. LetS= Spec(R). LetAbe the N´eron model ofAover S and letAl and Av be its generic and special fiber, that is the fibers lying over the generic point and the special pointv ∈Spec(Ov), respectively. In particularAl is isomorphic to A.
It is shown in [9, §6.5, Corollary 3] that A is also the N´eron model of A over Spec(Rsh), where Rsh is the strict henselization of R, with field of fractions lsh. The advantage of working over Spec(Rsh) is that the residue field ofRsh is separably closed.
Definition 1.16. Suppose the special fiberAv has irreducible components A0v, . . .Anv, wheren is a nonnegative integer and A0v is the connected com-ponent of the identity of Av. Thegroup of components Φv of Av is defined by
Φv :=Av/A0v.
The nonnegative integer cv := #Φv(l) is called the Tamagawa number of A/l. Furthermore, the identity component A0 of A is defined as the open subscheme ofA with generic fiber A and special fiberA0v. We defineA0 to be the subset ofA of points mapping to the connected componentA0v.
Note that A0v is always defined over l. Because of the group scheme structureA0 is a subgroup ofA and we have
Φv ∼=A(lsh)/A0(lsh) and
Φv(l)∼=
A(lsh)/A0(lsh)Gal(lsh/l)
∼=A(l)/A0(l).
The last isomorphism is not obvious, but follows from A0(lsh)Gal(lsh/l) = A0(l) and the vanishing ofH1 Gal(lsh/l), A0(lsh)
. The latter statement is part of the proof of [72, Chapter 1, Proposition 3.8].
IfP ∈ Alis anl-rational point on the generic fiber, then, by the universal property of the N´eron model, this point is the image of the generic point of Sunder a sectionσP :S −→ Aand the image of the special pointv∈S lies in one of the components of the special fiber. LetD∈Div(A)(l) be a prime divisor. We write its Zariski closure on Aas DA; this is a prime divisor on Aand if P does not lie in the support of D, then pulling this divisor back to S gives
σ∗P(DA) =i(D, P)(v) ∈Div(S) (1.3) for some well-defined integeri(D, P), because any divisor onSis an integral multiple of the special pointv. We calli(D, P) the intersection multiplicity of D and P at v. This construction can be extended to arbitrary D ∈ Div(A)(l) by linearity. In general this is not an intersection multiplicity in the usual sense, since the N´eron model might not be proper and hence one would need a completion satisfying certain properties in order to construct
a reasonable intersection theory on it. Such a completion is not known to exist in general, but see the proof of Proposition 2.14 below for the elliptic curve case. Also see [59, §12.3] for a discussion of this issue.
We can compute i(D, P) using the following observation: If DA is rep-resented by f ∈l(A) =l(Al) aroundσP(v), then we have
i(D, P) =v(f(P)), (1.4) and this does not depend on the choice of f. For the next theorem we specializeR further to the rings that we are interested in.
Theorem 1.17. (N´eron, Lang) Let kv be the completion of a number field or a one-dimensional function field at a non-archimedean place v with ring of integers Ov. Let A be an abelian variety defined over kv and let A be its N´eron model over Spec(Ov). Let D ∈Div(A)(kv) and let λD,v be a N´eron function associated withD and v.
For each component Ajv there is a constant γj(D)∈Q such that for all P ∈A(kv)\supp(D)
mapping to Ajv we have
λD,v(P) = Nv
nv(i(D, P) +γj(D)).
Proof. See [59, Chapter 11, Theorem 5.1].
The preceding theorem shows that the canonical height on an abelian variety is intimately related to intersection multiplicities on the correspon-ding N´eron models over the rings of integers of the completions. Indeed, N´eron’s original construction used these intersection multiplicities, mainly developed by N´eron himself, in a crucial way. It is possible to say more about the possible denominators ofγj(D); this is done by Lang in [59, Chapter 11, Theorem 5.2].