In a recent paper, Rosen and Silverman showed that Tate's conjecture on algebraic cycles implies a formula of Nagao, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. In this article, we extend their result to the case of elliptic threefolds. The main ingredients of our argument are a Shioda-Tate-like formula for elliptic threefolds, and a relation between the 'average' number of rational points on singular fibers and the Galois action on those fibers.

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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.