Torsion sections of Abelian fibrations

Abstract:

Let $K$ be a number field, and let $B$ be a smooth projective curve over $K$ of genus $\le 1$ such that $B(K)$ is infinite. Let $A$ be an Abelian variety defined over the function field $K(B)$. Suppose there are infinitely many non-trivial, pairwise disjoint extensions $L/K$ of bounded degree such that $B(L)$ is infinite and that $A_P(L)_\textrm {tor}\not =0$ for every point $P\in B(L)$ at which the specialization $A_P$ is smooth. We show that $A(K(B))_\textrm {tor}\not =0$. If $A/K(B)$ is not a constant Abelian variety over $K$, the extensions $L/K$ need not have bounded degree, and we can replace $B(K)$ being infinite by $B(K)\not =\emptyset$. This provides evidence in support of a question of Graber-Harris-Mazur-Starr on rational pseudo-sections of arithmetic surjective morphisms.