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

TL;DR

This paper investigates the Brauer-Manin obstruction for constant curves over global function fields, establishing its role as the only obstruction under certain genus conditions and using Frobenius descents to identify specific points.

Contribution

It demonstrates that Brauer-Manin is the sole obstruction to weak approximation and the Hasse principle when the genus of D is less than that of C, and introduces Frobenius descent techniques.

Findings

Brauer-Manin is the only obstruction under genus conditions.

Identifies points from non-constant maps using Frobenius descents.

Establishes conditions for weak approximation and Hasse principle.

Abstract

Let $\mathbb{F}$ be a finite field and $C,D$ smooth, geometrically irreducible proper curves over $\mathbb{F}$ and set $K = \mathbb{F}(D)$. We consider Brauer-Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve $C \otimes_\mathbb{F} K$. In particular, we show that Brauer-Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of $D$ is less than that of $C$. We also show that we can identify the points corresponding to non-constant maps $D \to C$ using Frobenius descents.