Natural bounded families of Néron-points for the additive group and other non-proper groups in Faltings' Néron-model construction

Faltings (Milan J. Math. 76, 2008) rebuilt the theory of Néron models over a discrete valuation ring using formal groups and invariant differential operators, and generalized the Néron extension property from all points to a chosen bounded, open family $E$ of Néron-points of a smooth separated group scheme $G_K$: his Theorem 1 attaches to any such family a unique smooth separated model over the base ring. For an abelian variety, taking all Néron-points recovers the classical Néron model; for the multiplicative group the valuation-zero points work; but for the additive group $\mathbb{G}_a$ the family of all Néron-points is unbounded, and the paper closes with the remark that there the choice of $E$ is 'very relevant', identifying no choice. The open problem is to determine natural or canonical choices of $E$ for such non-proper groups — the additive group being the critical case — and to analyze the integral models the construction then produces: their explicit structure and special fibres, their dependence on $E$, and the Néron-type mapping properties they retain. The classical existence question is settled negatively for groups containing $\mathbb{G}_a$ (Bosch–Lütkebohmert–Raynaud, Chapter 10; global case recently by Overkamp–Suzuki), which makes the bounded-family framework a genuine substitute.

Background

Let $V$ be a discrete valuation ring — the ring of integers of a local field such as $\mathbb{Z}_p$ or $k[[t]]$, i.e. a principal ideal domain with a single nonzero maximal ideal generated by a uniformizer $\pi$ — with fraction field $K$ and residue field $k$. Arithmetic geometry studies a smooth group scheme $G_K$ over $K$ (an algebraic group: an abelian variety, i.e. a smooth proper connected algebraic group; a torus $(\mathbb{G}_m)^n$; or the additive group $\mathbb{G}_a$, the algebraic group attached to $(K,+)$) through its integral models over $V$. The strongest model is the Néron model: a smooth, separated group scheme $G$ over $V$ with generic fibre $G_K$ such that every $K$-morphism from the generic fibre of a smooth $V$-scheme $S$ into $G_K$ extends uniquely to an $S$-morphism $S\to G$ (the Néron mapping property). Néron proved in 1964 that abelian varieties over local and global fields admit such models. The book of Bosch–Lütkebohmert–Raynaud (1990) developed the general theory — its construction passes through the smoothening process, which blows up a model until its rational points land in the smooth locus — and its Chapter 10, 'Néron Models of Not Necessarily Proper Algebraic Groups', treats groups that are not proper: relaxing 'of finite type' to 'locally of finite type' (Néron lft-models), it constructs Néron models of tori and proves the sharp local criterion that $G_K$ admits a Néron lft-model if and only if it does not contain $\mathbb{G}_a$; in particular no group containing $\mathbb{G}_a$ — and $\mathbb{G}_a$ itself — has a Néron model of any kind. Where the full Néron model fails one falls back on weak Néron models: smooth separated models, existing in general after a finite extension of $K$, whose points over the relevant valuation rings account for all rational points, but which satisfy no universal property and are far from unique. Faltings (2008) gave a streamlined construction of Néron models using invariant differential operators and formal groups — the formal completion of a smooth group scheme along its special fibre, carrying the structure of a group object in formal schemes — and extended the construction beyond the proper, commutative setting. His device is to impose the extension property not for all points but for a chosen family. A Néron-point of a $K$-scheme $X_K$ is a point of $X(K')$ where $V'\subset K'$ ranges over the valuation rings formally smooth over $V$ (same uniformizer, residue field separable over $k$); since base change of such points produces families of points rather than single points, products of Néron-points are themselves families. A family $E$ of Néron-points of a smooth separated $G_K$ is bounded if its elements extend to $V'$-points of some $V$-scheme $X$ of finite type with a map $X_K\to G_K$, and open if there is a smooth $V$-scheme $S$ with nonempty special fibre and a smooth dominant map $S_K\to G_K$ such that the reductions of the elements of $E$ that lift to $S$ are Zariski-dense in the special fibre $S_k$. Faltings' Theorem 1 states that for any bounded, open family $E$ of Néron-points, closed under inverses and products, there is a unique smooth separated $V$-group scheme extending $G_K$ such that every element of $E$ extends to a $V'$-point, these points' reductions are Zariski-dense in the special fibre, and the induced Néron-type mapping property holds for smooth $S$ whose liftable elements of $E$ have dense reductions. For an abelian variety $G_K$, taking $E$ to be the class of all Néron-points recovers the usual Néron model; for the multiplicative group $\mathbb{G}_m$ the Néron-points of valuation zero (the units) form a bounded family, and one may instead take the 1-units and similar subgroups. For the additive group, however, the family of all Néron-points is unbounded — elements of $K'$ of arbitrarily negative valuation cannot be absorbed by any finite-type $V$-scheme — so no 'all points' choice is available, and Faltings' paper closes with the remark that there 'the choice of $E$ is very relevant'. The paper identifies no choice for $\mathbb{G}_a$ and analyzes no resulting model; the same closing paragraph, and the non-commutative generality of Theorem 1, leave the analogous question open for general non-proper and non-commutative groups.

Problem Statement

Within Faltings' generalized construction (Milan J. Math. 76 (2008), Theorem 1), determine natural or canonical choices of the bounded, open family $E$ of Néron-points for smooth separated group schemes $G_K$ over the fraction field $K$ of a discrete valuation ring $V$ for which the family of all Néron-points is unbounded — the additive group $\mathbb{G}_a$ being the basic case singled out by the source — and analyze the integral models that the construction then produces. The problem asks to: (i) exhibit candidate families $E$ for $\mathbb{G}_a$ and, more generally, for other non-proper and possibly non-commutative groups, with proofs that they satisfy the source's defining conditions — boundedness via an explicit finite-type witness over $V$, openness with the density condition, and closure under inverses and base-change products; (ii) identify the resulting smooth separated $V$-models: their explicit structure, their special fibres, and how they depend on the choice of $E$; and (iii) establish to what extent they are useful integral models, that is, which Néron-type mapping properties they satisfy for which classes of test schemes, how they behave under extension of the base field $K$, and how they compare with weak Néron models on the one hand and with the classical non-existence of Néron models for groups containing $\mathbb{G}_a$ on the other. The problem further asks to make precise in what sense, if any, a choice of $E$ deserves to be called canonical — for example through functoriality in $V$, base-change compatibility, or a universal property characterizing it among bounded families.

The additive group is the basic case: exhibiting a candidate family $E$ for $\mathbb{G}_a$ over a stated class of discrete valuation rings, proving the source's defining conditions for it — boundedness via an explicit finite-type witness over $V$, openness with the density condition, and closure under inverses and base-change products — and identifying the smooth separated $V$-model Theorem 1 attaches to it together with that model's special fibre are part of the question. The remaining parts — other non-proper and possibly non-commutative groups, the comparison with weak Néron models, and the precise sense in which a choice of $E$ is canonical — also belong to the question.

References

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