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

TL;DR

This paper establishes that, under certain conditions, the Mordell-Weil rank of an abelian scheme over a surface remains constant when restricted to most vertical curves, linking it to recent results on Néron-Severi ranks.

Contribution

It reduces the problem of Mordell-Weil rank specialization over surfaces to the recently proved Néron-Severi rank specialization theorem in positive characteristic.

Findings

Mordell-Weil ranks are stable under specialization to most vertical curves.

The reduction uses the Shioda-Tate and Silverman theorems.

The result applies after a suitable blow-up of the base surface.

Abstract

Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields $k$ to the the specialization theorem for N\'eron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface $S$, for all vertical curves $S_x$ of a fibration $S \to U \subseteq \mathbf{P}^1_k$ with $x$ from the complement of a sparse subset of $|U|$, the Mordell-Weil rank of an abelian scheme over $S$ stays the same when restricted to $S_x$.