How do degeneracy loci constrain unlikely intersections in abelian schemes?

The finiteness claim holds outside one exceptional locus

The paper's main result concerns an abelian scheme A over a normal quasi-projective variety S, both defined over a number field K, and a positive-dimensional subvariety X of A defined over K. For each integer t>0, it proves that the set (X minus Xdeg(t)) intersected with A<t is finite, where A<t is the union of all flat group subschemes H of A whose relative dimension is less than t.

The exceptional set Xdeg(t) is the t-degeneracy locus of X. It is built from irreducible positive-dimensional subvarieties Y of X whose vertical defect is below t, where the vertical defect is the relative dimension of the relative weakly special closure of Y minus the dimension of Y. The paper calls X t-nondegenerate when Xdeg(t) is not all of X.

The immediate corollary is a nondensity statement: if X is t-nondegenerate, then X intersected with A<t is not Zariski dense in X. The theorem requires no simplicity or variation assumption on A over S, and it allows the group subschemes H to vary across the whole family rather than fixing one.

Positive-dimensional intersections are exactly what gets removed

The finiteness statement would be false without deleting the degeneracy locus. The paper notes that every positive-dimensional irreducible component Y of X intersected with a flat group subscheme H of relative dimension less than t satisfies a vertical defect bound: the relative dimension of the weakly special closure of Y minus the dimension of Y is at most the relative dimension of H minus the dimension of Y, which is less than t.

Such a component therefore lies inside Xdeg(t) by construction. This is why the theorem is stated for the complement of Xdeg(t) rather than for all of X, and it explains why the result is a finiteness statement about isolated points rather than about higher-dimensional pieces.

The paper also distinguishes A<t from the larger locus A(<t), which is the union of all algebraic subgroups of dimension less than t in the geometric fibers. Since A<t is contained in A(<t), the theorem is a statement about flat group subschemes specifically, and the authors present it as the flat-subgroup part of the codimension range predicted by Pink's conjecture.

Bounded height alone is not enough for finiteness

The proof starts from a bounded height theorem of the third-named author. For a t-nondegenerate subvariety X defined over K, the intersection of X minus Xdeg(t) with A≤t has bounded total height, where the total height is the sum of the fiberwise Neron-Tate height and a Weil height on the base. The paper uses this bound as its arithmetic starting point.

Bounded height does not by itself give finiteness, because the points involved may have arbitrarily large degree over K. The authors therefore relate the field degree of a point to the complexity of the subgroup relations it satisfies. For a point x in a fiber, they define its Gamma-special closure as the common kernel of the endomorphisms in Gamma that annihilate x, and they measure its complexity by the degree of that closure, denoted Delta_Gamma(x).

Theorem 5.8 gives an explicit upper bound for this complexity in terms of field degree, heights, and the endomorphism lattice, combining geometry of numbers with estimates of Gaudron-Remond. For a fixed family and a fixed total height bound, this yields a large-Galois-orbit estimate of the form [K(x):Q] at least c times Delta_Gamma(x) to the power epsilon, with constants c and epsilon greater than zero.

Counting plus Ax-Schanuel pushes large-complexity points into the locus

The proof follows the Pila-Zannier strategy, combining o-minimal point counting with Diophantine estimates. In Betti coordinates, subgroup relations can be encoded by rational data whose height is bounded polynomially in Delta_Gamma(x). If a point of the intersection had large complexity, its Galois orbit would supply enough such data to apply the Habegger-Pila refinement of the Pila-Wilkie counting theorem and produce a nonconstant analytic arc with the required semialgebraic property.

Mixed Ax-Schanuel, in the form of Proposition 4.3 in the paper, then implies that this arc lies inside the degeneracy locus. That contradicts the choice of the point, which was taken outside Xdeg(t).

The contradiction bounds the complexity of every point in the relevant intersection. With complexity bounded, only finitely many flat group subschemes remain to be considered, and their positive-dimensional intersections with X are already contained in the degeneracy locus. This is what yields the finiteness in the main theorem.

The theorem recovers several known cases but not all of them

When S is a point, the theorem recovers the finiteness statement of Habegger-Pila and its nondensity consequence. The paper is explicit that for curves in a fixed abelian variety, its nondegeneracy hypothesis requires the curve not to be contained in any translate of a proper abelian subvariety, which is stronger than the hypothesis in the earlier result, so the theorem does not recover that result in full.

When g is at least 2 and X is a curve, taking t equal to g minus 1 gives finiteness of the intersection with flat group subschemes of relative dimension at most g minus 2, provided X is (g minus 1)-nondegenerate. If S is a curve, X dominates S, and A over S has no fixed part, the geometric hypothesis that X is not contained in a strict group subscheme even after a finite base change implies nondegeneracy, so the theorem recovers the earlier curve result in that case.

For t equal to 1, the locus A<1 is the fiberwise torsion locus. The theorem then gives finiteness of torsion points outside Xdeg(1), which implies the degeneracy statement for the universal family in Gao-Habegger. Together with an induction argument and the usual reductions to the universal family, this yields the relative Manin-Mumford conjecture over Q.

A relative Mordell-Lang conjecture follows from Pink's conjecture

The first application concerns a finitely generated subgroup Lambda of the group of sections A(S). The paper defines the divisible hull of the specialization of Lambda in each fiber and proposes a relative Mordell-Lang conjecture governing the locus of points in X(K-bar) that lie in that divisible hull as the base point varies.

The conjecture assumes that X dominates S, has geometrically irreducible generic fiber, and satisfies a transversality condition: the geometric generic fiber of X is not contained in any translate of a proper abelian subvariety by a point of the divisible hull of Lambda at the generic point. Under these conditions it predicts nondensity whenever the dimension of X is less than the relative dimension of A.

Proposition 7.2 shows that Pink's Conjecture 1.1 implies this relative Mordell-Lang conjecture. Proposition 7.4 then proves the conjecture for section images in powers of an abelian scheme, under the additional hypothesis that every nontrivial simple abelian subvariety of the geometric generic fiber has maximal variation, meaning its moduli map is generically finite onto its image, and under the dimension condition that dim S is less than n times g.

Specialization is injective outside a proper closed subset

The second application studies the injectivity of specialization of Lambda. The paper defines Sigma(Lambda) as the set of base points s in S(K) where the specialization map from Lambda to the fiber A_s is not injective, and proves that Sigma(Lambda) is not Zariski dense in S.

The hypotheses are that every nontrivial simple abelian subvariety of the geometric generic fiber has maximal variation and dimension greater than dim S. The paper notes that the maximal variation assumption on the simple components is equivalent to A having no fixed part when S is a curve, and that the dimension assumption is what makes the intersection unlikely.

The proof reduces the failure of injectivity to a point of a section lying in the kernel of a nonzero homomorphism between abelian subschemes, and then applies the main finiteness theorem. The paper presents this as complementing bounded height results for specialization by establishing Zariski nondensity under suitable variation and dimension hypotheses. Over a curve, the exceptional set is finite under these hypotheses.

The t-nondegeneracy check is the decisive condition

The most consequential condition in this paper is t-nondegeneracy of X, because it is what separates the finiteness conclusion from the trivial case where the intersection is already positive-dimensional. The theorem's finiteness statement is about the complement of Xdeg(t), and the nondensity corollary only applies when Xdeg(t) is a proper closed subset of X.

A second condition to track is the distinction between A<t and A(<t). The theorem controls flat group subschemes of relative dimension less than t, not all algebraic subgroups of dimension less than t in the geometric fibers, so a reader comparing this result with Pink's conjecture should keep the flat-subgroup restriction in view.

For the applications, the decisive conditions are different: the relative Mordell-Lang result for section images requires maximal variation of every nontrivial simple factor of the geometric generic fiber and the inequality dim S less than n times g, while the injectivity result requires maximal variation together with the dimension of each such factor exceeding dim S. These are the thresholds that determine whether the paper's conclusions apply to a given family.

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