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The Brauer-Manin obstruction for nonisotrivial curves over global function fields

Brendan Creutz, José Felipe Voloch

TL;DR

The article proves that nonisotrivial curves of genus at least 2 over global function fields satisfy X({\mathbb A}_K)^{Br}=X(K), extending prior work by removing ambient abelian-variety hypotheses. The authors develop a Mordell-Lang–type proposition and employ Rößler's p^n-isogeny to control descent, linking adelic intersections to rational points via a finite-subscheme reduction. This yields a function-field analogue of Scharaschkin–Skoorobogatov-type obstructions for curves and highlights a path toward higher-dimensional generalizations, with isotrivial cases remaining open. The work integrates adelic methods, descent theory, and isogeny techniques to establish a precise obstruction–rational-point correspondence for the problem.

Abstract

We prove that the set of rational points on a nonisotrivial curves of genus at least 2 over a global function field is equal to the set of adelic points cut out by the Brauer-Manin obstruction.

The Brauer-Manin obstruction for nonisotrivial curves over global function fields

TL;DR

The article proves that nonisotrivial curves of genus at least 2 over global function fields satisfy X({\mathbb A}_K)^{Br}=X(K), extending prior work by removing ambient abelian-variety hypotheses. The authors develop a Mordell-Lang–type proposition and employ Rößler's p^n-isogeny to control descent, linking adelic intersections to rational points via a finite-subscheme reduction. This yields a function-field analogue of Scharaschkin–Skoorobogatov-type obstructions for curves and highlights a path toward higher-dimensional generalizations, with isotrivial cases remaining open. The work integrates adelic methods, descent theory, and isogeny techniques to establish a precise obstruction–rational-point correspondence for the problem.

Abstract

We prove that the set of rational points on a nonisotrivial curves of genus at least 2 over a global function field is equal to the set of adelic points cut out by the Brauer-Manin obstruction.
Paper Structure (5 sections, 9 theorems, 19 equations)

This paper contains 5 sections, 9 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Zariski dense adelic points surviving $p^{\infty}$-descent
  3. Rational points on finite subschemes of abelian varieties
  4. Proofs of the theorems
  5. On abelian varieties with an infinite group of separable $p^\infty$-torsion points (by Damian Rössler)

Key Result

Theorem 1.1

Let $X/K$ be a smooth projective curve of genus at least $2$ over a global function field $K$. If $X$ is not isotrivial, then $X({\mathbb A}_K)^{\mathop{\mathrm{Br}}\nolimits} = X(K)\,.$

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • ...and 14 more