Néron Models 1


Let’s start with the exciting news that today I broke 100,000 views!! Thanks to all my long-time and new subscribers. I’ve decided to talk about Néron models. The other idea was just too overwhelmingly big that I’d probably use that as an excuse not to post. Let’s start with the general idea of Néron models. Suppose you start with some abelian variety, {A_K}, over a number field {K}. You could view this as an abelian variety that is defined generically over the ring of integers of {K}, say {R}. From this you can “spread out” the variety and get a smooth scheme over an open subset of {S=Spec R}. You could define the spread out {A} over all of {Spec R}, but it won’t be smooth because it will have bad reduction at finitely many places.

The definition of Néron model will not require that {A_K} be an abelian variety. For this series of posts we’ll probably keep switching between {A_K} and {X_K} for the notation. It essentially won’t make any difference whether or not the scheme has a group structure. The reason for the confusion is that most applications right now have to do with abelian varieties.

The idea of a Néron model is to construct an honest smooth, separated, finite type scheme {X} over {Spec R} which will be canonical in some sense and also has the property that for any smooth {S}-scheme {Y} and any morphism {u_K: Y_K \rightarrow X_K} there is a unique {S}-morphism {u:Y\rightarrow X} extending {u_K} (the Néron mapping property).

For the rest of the post we’ll just describe properties we want the Néron model to have along with some related remarks. The generality that we will mostly work with is to let {S} be an arbitrary Dedekind scheme with field of fractions {K}. Given any scheme {X_K} over {Spec K} we call a scheme {Y} over {S} an {S}-model for {X_K} if {Y\otimes K\simeq X_K}. Without specifying more properties, we clearly won’t have a unique or canonical model since we can just change {Y} in sufficiently high codimension away from the generic fiber and get non-isomorphic {S}-models.

One property we will want our Néron model {X} to satisfy is that {X(R^{sh})\rightarrow X_K(K^{sh})} is surjective. One way to think about this property is that every rational point on our starting variety has some integral point on the model that specializes to it. Later we might relate this to a notion of boundedness.

We can say that Néron models are “minimal” in the sense that if you have any other {S}-model {Y}, then there is a unique morphism {Y\rightarrow X} restricting to the identity on the generic fiber. Unravelling this further we find that the Néron model is canonical in the sense that {X} is uniquely determined up to canonical isomorphism by {X_K}. One interesting question that you should be asking is if we are given some arbitrary {X/S}, can we tell if it is a Néron model of its generic fiber? A quick condition to check is that this happens if and if only for all closed points {s\in S}, the {\mathcal{O}_{S, s}}-scheme {X\times_S Spec \mathcal{O}_{S, s}} is a Néron model of its generic fiber.

Going back to the abelian variety examples. If we build a Néron model of {A_K}, the group scheme structure uniquely extends to the model. This is further reason to not worry about abelian varieties in the general theory. The previous paragraph simplifies greatly when we work with a group scheme. If {A/S} is an abelian scheme, then it is always a Néron model of its generic fiber. This just follows from the criterion given and Weil’s extension theorem.

The last thing to point out today is that Néron models are often not the obvious choice of model. For example, take {X_K=\mathbb{P}^1_K}. The variety {X=\mathbb{P}^1_S} is a smooth, separated, finite type {S}-model of {X_K}, so we might hope that it is the Néron model, but it isn’t. It even satisfies this extension property for étale points (a consequence of the Néron mapping property, but not equivalent to it). Next post we’ll have a technique to actually check this isn’t the Néron model, but for now just take it as a word of caution.

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