Adelic Mordell-Lang and the Brauer-Manin obstruction

TL;DR

The paper analyzes how rational points on a subvariety $X$ of an abelian variety $A$ over global fields relate to their adelic and Brauer-set images under the nonisotriviality hypothesis. It develops and connects Selmer-group machinery, Brauer-Manin obstructions, and adelic Mordell–Lang-type results to show that, for function fields with no isotrivial quotient, $ar{X(k)}=X({f A}_k)^{ m Br}$ and, in particular, $X(k)$ is dense in the Brauer set when ${ m Sh}(A)_{ m div}=0$. The paper introduces and exploits Rosser exceptional schemes and Wisson’s adelic Mordell–Lang framework to prove a finite union of cosets controls adelic intersection data, with extensions to totally imaginary number fields under an AML conjecture. Together, these results provide new instances where the Brauer-Manin obstruction fully accounts for the failure of weak approximation and the density of rational points, advancing the adelic Mordell–Lang program in the function-field setting.

Abstract

Let $X$ be a closed subvariety of an abelian variety $A$ over a global function field $k$ such that the base change of $A$ to an algebraic closure does not have any positive dimensional isotrivial quotient. We prove that every adelic point on $X$ which is the limit of a sequence of $k$-rational points on $A$ is a limit of $k$-rational points on $X$. Assuming finiteness of the Tate-Shafarevich group of $A$, this implies that the rational points on $X$ are dense in the Brauer set of $X$. Similar results are obtained over totally imaginary number fields, conditionally on an adelic Mordell-Lang conjecture.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof