Quasi-compactness of Néron models, and an application to torsion points

We prove that Néron models of Jacobians of generically-smooth nodal curves over bases of arbitrary dimension are quasi-compact (hence of finite type) whenever they exist. We give a simple application to the orders of torsion subgroups of Jacobians over number fields.


Introduction
If S is a regular scheme, U ⊆ S is dense open, and A/U is an abelian scheme, then a Néron model for A/S can be defined by exactly the same universal property as in the case where S has dimension 1. We do not impose a priori that the Néron model should be of finite type, and we allow the Néron model to be an algebraic space (rather than just a scheme) for flexibility. We investigate this in detail in [8], giving necessary and sufficient conditions for the existence of Néron models in the case of Jacobians of nodal curves. In this short note we prove that, in the setting of Jacobians of nodal curves, a Néron model is quasi-compact (and hence of finite type) over S whenever one exists (Theorem 2.1).
Note that Néron models of non-proper algebraic groups (such as G m ) need not be quasi-compact. Moreover, in [2, §10.1,11] an example (due to Oesterlé) is given of a Néron model over a dedekind scheme all of whose fibres are quasicompact, but which is not itself quasi-compact. Thus in general the question of quasi-compactness of Néron models can be somewhat delicate.
We give an application to controlling the orders of torsion points on abelian varieties. Recall that by [4] the uniform boundedness conjecture for Jacobians of curves is equivalent to the same conjecture for all abelian varieties. By considering the universal case, the uniform boundedness conjecture is equivalent to Conjecture 1.1. (Uniform boundedness conjecture) Let U be a scheme of finite type over Q and C/U a smooth proper curve. Let σ ∈ J (U ) be a section of the Jacobian. Let d ≥ 1 be an integer. Then there exists an integer B such that for every point u ∈ U (Q) with [κ(u) : Q] ≤ d, the order of the point σ (u) ∈ J u is either infinite or at most B.
We will show (Corollary 3.5) that this conjecture holds if there exists a compactification U → S over which C has a proper regular model with at-worst nodal singularities, and over which J has a Néron model. The first two conditions are relatively mild, especially since it is actually enough to work up to alterations (cf. [5]), but the assumption that a Néron model should exist is very strong. For the sake of those readers unfamiliar with the theory developed in [8] This can be viewed as a common generalisation (in the case of Jacobians) of the following standard results: Proposition 1.3 is easy to prove and is well-known; Proposition 1.3 follows for example from [11]. Other examples are given in Sect. 3.1.
Neither Corollary 1.2 nor the stronger form Corollary 3.5 is very useful for proving that the full uniform boundedness conjecture holds for a given family of Jacobians; a Néron model exists for the tautological curve over a family of curves C/S if and only if C/S is of compact type, in which case the uniform boundedness conjecture is in any case easy (cf. Corollary 3.4).
In [9] we consider related questions where we replace torsion points with points of 'small height'. However, the results of [loc.cit] are more restrictive since we need to make an additional technical positivity assumption on the base S of our family. The techniques used to prove those results are very different from what we do here, though [loc.cit] does use the quasi-compactness result proven in this paper.
Many thanks to Owen Biesel, Maarten Derickx, Bas Edixhoven, Wojciech Gajda, Ariyan Javanpeykar, Robin de Jong and Pierre Parent for helpful comments and discussions. The author would also like to thank the anonymous referee for a very rapid report suggesting a number of improvements. Finally, the author is very grateful to an anonymous referee at Crelle for pointing out a gaping hole in an earlier proof given in [8] of the quasi-compactness of such Néron models.

The Néron model is of finite type
Before giving the proof of our main theorem we briefly recall some definitions we need from [7,8]. In what follows, S is a scheme. Details can be found in the above references.
A nodal curve over S is a proper, flat, finitely presented morphism, all of whose geometric fibres are reduced, connected, of dimension 1, and have at worst ordinary double point singularities. If C/S is a nodal curve then we write Pic [0] C/S for the subspace of Pic C/S consisting of line bundles which have total degree zero on every fibre. If s ∈ S is a point then a non-degenerate trait through s is a morphism f : T → S from the spectrum T of a discrete valuation ring, sending the closed point of T to s, and such that f * C is smooth over the generic point of T .
We say a nodal curve C/S is quasisplit if the morphism Sing(C/S) → S is an immersion Zariski-locally on the source (for example, a disjoint union of closed immersions), and if for every field-valued fibre C k of C/S, every irreducible component of C k is geometrically irreducible.
Suppose we are given C/S a quasisplit nodal curve and s ∈ S a point. Then we write s for the dual graph of C/S at s-this makes sense because C/S is quasisplit and so all the singular points are rational points, and all the irreducible components are geometrically irreducible. Assume that C/S is smooth over a schematically dense open of S. If we are also given a non-smooth point c in the fibre over s, then there exists an element α ∈ O S,s and an isomorphism of completed étale local rings (after choosing compatible geometric points lying over c and s) This element α is not unique, but the ideal it generates in O S,s is unique. We label the edge of the graph s corresponding to c with the ideal αO S,s . In this way the edges of s can be labelled by monogenic ideals of O S,s . If η is another point of S with s ∈ {η} then we get a specialisation map sp : s → η on the dual graphs, which contracts exactly those edges in s whose labels generate the unit ideal in O S,η . If an edge e of s has label , then the label on the corresponding edge of η is given by O S,η . Proof. The Néron model is smooth and hence locally of finite type; we need to prove that it is quasi-compact over S.
The existence of the Néron model implies by [8] that the Néron model coincides canonically with the quotient of Pic [0] C/S by the closure of the unit sectionē, and the latter is flat (even étale) over S. Given an integer n ∈ N, we write Pic C/S → N . If we can show that this composite is surjective for some n then we are done. We may assume without loss of generality that S is noetherian, since every regular scheme is locally noetherian, and quasi-compactness is local on the target. Then we are done if we can make a constructible function n : S → N such that for all s ∈ S, the composite Pic We are done if we can find a non-empty open subset V → {s 0 } and an integer n ∈ N such that for all s ∈ V , the composite Pic Step 2: Graphs and test curves.
After perhaps replacing S by an étale cover, we may assume that C/S is quasisplit. By [7, lemma 6.3] there exists an open subset V → {s 0 } such that for all s ∈ V , the specialisation map sp : s → s 0 is an isomorphism on the underlying graphs.

Claim: After shrinking V , there exists an integer m such that for all s ∈ V there exists a non-degenerate trait f s : T s → S through s such that for every edge e of s , we have
|ord T s f * s label(e)| ≤ m.
Two things remain to complete the proof: we must prove the claim, and deduce the theorem from the claim.
Step 3: Proving the claim. Replacing S by an étale cover (and s 0 by a point lying over it) we can arrange the following: -S is affine (say S = Spec A); -There exist generators a 1 , . . . , a n of the prime ideal s 0 ⊆ A such that for each 0 ≤ i ≤ n, the closed subscheme V (a 1 , . . . , a i ) ⊆ S is connected and regular, is of codimension i, and is not contained in the closed subscheme of S over which C is not smooth. Let X be the closed subscheme of S cut out by a 1 . . . , a n−1 . Then X is regular and connected, and the image x of s 0 in X is of codimension 1, cut out by the image a ∈ O X (X ) of a n . For each edge e of , write label(e) ∈ O S,x for its label. After shrinking S, we may assume that all these labels lie in O S (S). The images of the labels on X lie in O X (X ), and so after perhaps shrinking X we can assume that every label is equal (up to multiplication by a unit in O X (X )) to some power of a. Set V := V (a) ⊆ X . Fix v ∈ V , and write m v for the maximal ideal of the local ring at v. By the regularity assumptions we see that a ∈ m v and a / ∈ m 2 v . Hence we can find a set of elements r 1 , . . . , r d ∈ m v such that the image of a, r 1 , . . . , r d in m v /m 2 v are a basis as an O X,v /m v -vector space. Then define T v to be the subscheme V (r 1 , . . . , r d ) of Spec O X,v , and it is clear that a pulls back to a uniformiser in the trait T v , so we have Step 4: Deducing the theorem from the claim.
We know the formation of Pic [0] C/S commutes with base change. As remarked above,ē is flat over S by our assumption that the Néron model exists. Because of this flatness, it follows that formation of the closure of the unit section also commutes with base-change. Now let us fix for each s ∈ V a non-degenerate trait f s : T s → S through s as in the statement of the claim. Then the thicknesses of the singularities of C T s /T s are bounded in absolute value by m as s runs over V (note that the graphs of the central fibres are the same for all s ∈ V ). By [6, §2] the multidegrees of points in the closure of the unit section can be described purely in terms of the combinatorics of the dual graph. Since there are only finitely many possibilities for this dual graph, we find an integer n such that for all s ∈ V , the composite

Consequences for torsion points
Given a field k, by a variety over k we mean a separated k-scheme of finite type. We fix a number field K and an algebraic closureK of K . We write κ( p) for the residue field of a point p. If X/K is a variety and d ∈ Z ≥1 then we write X (K ) ≤d for the set of x ∈ X (K ) with [κ(x) : K ] ≤ d.
Definition 3.1. Let S/K be a variety, and A/S an abelian scheme.

We say the uniform boundedness conjecture holds for A/S if for all d ∈ Z ≥1
there exists B ∈ Z such that for all s ∈ S(K ) and all torsion points p ∈ A(K ) ≤d s , the point p has order at most B. 2. Given a section σ ∈ A(S), we say the uniform boundedness conjecture holds for the pair (A/S, σ ) if for all d ∈ Z ≥1 there exists B ∈ Z such that for all p ∈ S(K ) ≤d , the point σ ( p) either has infinite order or has order at most B.
If the uniform boundedness conjecture holds for A/S then it clearly holds for the pair (A/S, σ ) for all σ . By considering the case of the universal PPAV, we deduce that if the uniform boundedness conjecture holds for all pairs (A/S, σ ) then the uniform boundedness conjecture itself (Conjecture 1.1) holds.

Lemma 3.2.
Fix an integer g ≥ 0 and a prime power q. Then there exists an integer b = b(g, q) such that for every connected commutative finite-type group scheme G/F q of dimension g we have #G(F q ) ≤ b.
Proof. Since F q is perfect, the scheme G red is a subgroup scheme and contains all the field-valued points, so we may assume G is reduced and hence smooth. Again using that F q is perfect, we can apply Chevalley's theorem to write an extension where A is abelian, T is a torus and U is connected and unipotent. We know U is isomorphic (as a scheme over F q ) to A n F q for some n ≤ g by [10,Remark A.3], so we have uniform bounds on the sizes of A(F q ), T (F q ) and U (F q ), from which the result is immediate. Proof. To simplify the notation we treat the case K = Q and d = 1; the general case is very similar. We begin by observing that we can 'spread out' the proper scheme S, the group algebraic space A and the section σ over Z[1/N ] for some sufficiently divisible integer N -we use the same letters for the 'spread out' objects. Let p be a prime number not dividing N , and let b = b(dim A/S, p) be the bound from Lemma 3.2, noting that a finite-type group algebraic space over a field is a group scheme by [1]. Suppose we are given s ∈ S(K ) (with unique extension s ∈ S(Z[1/N ]) since S is proper over Z[1/N ]) such that σ s is torsion. For some n the point τ := p n σ s is torsion of order prime to p. Then the subgroup algebraic space of As generated by τ is étale over Z p , and so the order of the torsion point τ is bounded above by b (independent of s ∈ S(K )). In this way we control the prime-top part of the order, and by considering another prime l we can control the whole order.
In the case of abelian varieties, we immediately obtain By considering the diagonal section of the tautological family of abelian varieties A × S A over A, we deduce that the uniform boundedness conjecture holds for an abelian scheme over a proper smooth base, i.e. when U = S (Proposition 1.4).
The proper smooth case is rather trivial, and it is not a priori clear how to construct interesting examples for Corollary 3.4. In the case of Jacobians of nodal curves the situation becomes much better; we understand exactly when Néron models exist by [8], and by Theorem 2.1 we know that they are always of finite type, so any section of the jacobian family extends to the identity component of the Néron model after taking some finite multiple (which is harmless for the arguments). We obtain Corollary 3.5. Let S be a proper scheme over K and C/S a regular family of nodal curves, smooth over some dense open U ⊆ S. Let σ ∈ J (U ) be a section of the jacobian family. If J/U admits a Néron model over S then the uniform boundedness conjecture holds for (J/U, σ ).

Examples
We finish by giving some examples where this result can be applied. Recall from [8] that the jacobian J admits a Néron model if and only if the curve C/S is aligned; in other words that for all geometric points s of S, the labelled dual graph s described in Sect. 2 has the property that for every circuit γ in s , and for every pair of edges e 1 , e 2 appearing in γ , the labels of e 1 and e 2 satisfy a multiplicative relation of the form label(e 1 ) n 1 = label(e 2 ) n 2 for some positive integers n 1 and n 2 .
Thus we see that J admits a Néron model over S (and so Corollary 3.5 applies) if any of the following hold (note that in each case S must be proper in order to apply Corollary 3.5).
X/M 1,2 does have treelike fibres, but when we resolve the singularities this breaks down, and indeed M 1,3 over M 1,2 does not have treelike fibres and its jacobian does not admit a Néron model. Similar considerations can be used to see that our results cannot be used to recover those of [3].
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