A specialisation theorem for Lang-N\'eron groups

TL;DR

This paper proves a specialization theorem for Lang-Néron groups on certain algebraic varieties, showing that for most hyperplane sections, the associated abelian variety has good reduction and injective specialization, advancing understanding of Néron's theorem.

Contribution

It establishes a new specialization theorem for Lang-Néron groups on polarized smooth projective varieties, confirming a conjecture and providing insights into Néron's theorem.

Findings

Existence of a dense open set of hyperplane sections with good reduction

Injectivity of the specialization homomorphism for Lang-Néron groups at these sections

Implications for height pairings and Néron's specialization theorem

Abstract

We show that, for a polarised smooth projective variety $B \hookrightarrow \mathbb{P}^n_k$ of dimension $\geq 2$ over an infinite field $k$ and an abelian variety $A$ over the function field of $B$, there exists a dense Zariski open set of smooth geometrically connected hyperplane sections $h$ of $B$ such that $A$ has good reduction at $h$ and the specialisation homomorphism of Lang-N\'eron groups at $h$ is injective (up to a finite $p$-group in positive characteristic $p$). This gives a positive answer to a conjecture of the first author, which is used to deduce a negative definiteness result on his refined height pairing. This also sheds a new light on N\'eron's specialisation theorem.