On the Specialization Theorem for Abelian Varieties

Abstract

In this paper, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's specialization theorem for families of Abelian varieties over K, where K is any field finitely generated over Q.Let k be a number field, and let π : A → B be a proper flat morphism of smooth projective varieties over k such that the generic fiber A η is an Abelian variety defined over k (B). For almost all absolutely irreducible divisors D/k on B, A D := A× B D is also a flat family of Abelian varieties, and our goal is to compare the Mordell-Weil rank of A η with the Mordell-Weil rank of the generic fiber of A D . In fact, if we fix a projective embedding of A, B into P N , and an integer M > 0, then we can show that for all but finitely many divisors D of degree less than M , the rank of the generic fiber of A D is at least the rank of A η .This result is an amusing example of the height machine in action. The main issue is to rephrase the problem in terms of Abelian varieties over function fields, and to define the 'right' height functions; the proofs then follow verbatim as in [5].Assume now that we also have a proper, flat morphism f : B → X with generic fiber a smooth, irreducible curve C defined over K := k(X ). Composition gives a flat morphism g := f • π : A → X whose generic fiber is a smooth, irreducible variety A defined over K, and by base extension we have a flat morphism ρ : A → C, whose generic fiber is A η . Observe that in this setting, divisors D ∈ Div(B) such that f (D) = X correspond to points on C. The next proposition (based on a classical geometric argument; see [2, Proposition 5.1]) shows that, up to replacing B and A by birationally equivalent varieties, we can always reduce to this situation.

Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves

“…The theorem for k of characteristic 0 is [Waz06, Theorem 1 and the text before Proposition 1]. We merely describe the necessary changes when k is of positive characteristic: We only have to see that we have the 'height machine' for the arithmetic and geometric height in [Waz06] (which generalizes Silverman's specialization theorem [Sil83]). We sketch the proof: As almost all curves on A can be realized as horizontal curves with respect to finitely many fibrations of A over curves (possibly after blowing up A) [Waz06, Proposition 1], suppose we are in such a situation: Assume A is fibered as ρ : A → C over a curve C. Fix a line bundle L on A and denote by D ρ its restriction to the generic fiber A ρ of ρ.…”

“…Silverman's specialization theorem [Sil83] states that for a family A of abelian varieties fibered over a curve C over a number field K, the specialization homomorphism from the generic fiber A (K(C)) to a special fiber A (κ(x)) is injective for x ∈ C(K) outside a subset of bounded height. (The specialization theorem has been extended to finitely generated fields in characteristic 0 by [Waz06].) In particular, if one can construct such a family with generic Mordell-Weil rank r, one gets infinitely many specializations over number fields of rank at least r.…”

Boundedness in families with applications to arithmetic hyperbolicity

“…Let $$\lbrace \sigma _1, \ldots , \sigma _{r}\rbrace$$ be a basis for the free part of $$\mathcal {X}(\mathcal {C}) = \mathcal {X}_{K(\mathcal {C})}(K(\mathcal {C}))$$. Since $$L$$ is a finitely generated field over $$\mathbb {Q}$$, by Silverman's specialisation theorem [77, Theorem 1], there is a finite field extension $$L^{\prime}/L$$ contained in $$k$$ and a point $$c \in \mathcal {C}(L^{\prime}) \subset C(k)$$ such that $$\sigma _{1}(c), \ldots , \sigma _{r}(c) \in X_c(k)$$ are still independent. Let $$c \in C(k)$$ be such a point.…”

Specialization of Mordell–Weil ranks of abelian schemes over surfaces to curves

“…The theorem for k of characteristic 0 is [Waz06, Theorem 1 and the text before Proposition 1]. We merely describe the necessary changes when k is of positive characteristic: We only have to see that we have the 'height machine' for the arithmetic and geometric height in [Waz06] (which generalizes Silverman's specialization theorem [Sil83]).…”