Finite descent obstructions and rational points on curves

Abstract

Let $k$ be a number field and $X$ a smooth projective $k$-variety. In this paper, we study the information obtainable from descent via torsors under finite $k$-group schemes on the location of the $k$-rational points on $X$ within the adelic points. Our main result is that if a curve $C∕k$ maps nontrivially into an abelian variety $A∕k$ such that $A\left(k\right)$ is finite and $\text{ Ш}\left(k,A\right)$ has no nontrivial divisible element, then the information coming from finite abelian descent cuts out precisely the rational points of $C$. We conjecture that this is the case for all curves of genus at least 2. We relate finite descent obstructions to the Brauer–Manin obstruction; in particular, we prove that on curves, the Brauer set equals the set cut out by finite abelian descent. Our conjecture therefore implies that the Brauer–Manin obstruction against rational points is the only one on curves.