Lower bounds for the rank of families of abelian varieties under base change

Abstract

We consider the following question : given a family of abelian varieties A over a curve B defined over a number field k, how does the rank of the Mordell-Weil group of the fibres At(k) vary? A specialisation theorem of Silverman guarantees that, for almost all t in C(k), the rank of the fibre is at least the generic rank, that is the rank of A(k(B)). When the base curve B is rational, we show, at least in many cases and under some geometric conditions, that there are infinitely many fibres for which the rank is larger than the generic rank. This paper is a sequel to a paper of the second author [Sa12] where the case of elliptic surfaces is treated.

Rank jumps on elliptic surfaces and the Hilbert property

“…Given Silverman's result, a natural question is whether one can construct infinitely many fibres whose rank is greater than the generic rank. This has been achieved in various cases when X is unirational or a K3 surface [1,18,17,8].…”

Point générique et saut du rang du groupe de Mordell-Weil

“…(iv) Dans [7], M. Hindry et C. Salgado étendent un certain nombre des résultats de [15] au cas des familles de variétés abéliennes sur la droite projective. Leur théorème 1.4 est maintenant un cas particulier du théorème 3.3 ci-dessus.…”

“…Dans la version initiale du présent article, je m'étais limité à supposer W → X dominant. La généralisation au cas où π : g(W ) ad → U est dominant de dimension relative au moins égale à 1 a été suggérée par une remarque dans [7].…”

“…• Certains résultats de Billard [1], Salgado [15], Hindry-Salgado [7], Loughran-Salgado [9] (voir les remarques 3.6).…”

Rank jumps on elliptic surfaces and the Hilbert property

“…Given Silverman's result, a natural question is whether one can construct infinitely many fibres whose rank is greater than than the generic rank. This question has been studied in various cases when X is unirational or a K3 surface [2,16,17,8].…”