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(k) is finite and X(k, A) 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.

Conditional results on the birational section conjecture over small number fields

“…Thus, it follows immediately from [14], Theorem 8.2, together with condition (2), that Z(k) = ∅, hence also X(k) = ∅. This completes the proof of the implication (2) ⇒ (1), hence also of Theorem 4.1 in the case where s is a locally geometric pro-C Galois section.…”

“…[1], Theorem 2.1]. Next, let us recall that, in [4], Harari and Stix proved, as a consequence of results obtained by Stoll in [14], that if there exist an abelian variety A over k and a nonconstant morphism X → A over k such that both the Mordell-Weil group and the Shafarevich-Tate group of A/k are finite, then any birational Galois section of X/k is geometric [cf. [4], Theorem 17].…”

“…[14], Definition 5.4, (3)] that the [uniquely determined] k p -valued points of X associated to s -where p ranges over nonarchimedean primes of kform a part of an element of X(A k ) f-ab…”

“…Remark 4.2.1. As in the case of [4], Theorem 17, one may apply Corollary 4.2 to obtain some examples of projective smooth curves over number fields for which any prosolvable birational Galois section [i.e., any pro-C birational Galois section in the case where C consists of all finite solvable groups] is geometric [cf., e.g., the discussions in [4], Remark 18, (1); [14] (i) We shall say that s is cuspidal if the image of s is contained in a decomposition subgroup of Π C X/k associated to a cusp of X/k. (ii) We shall say that s is unramified almost everywhere if the composite Proof.…”

Rational points on curves

“…Still, we can formulate the following conjecture. In [52], we argue that there are good reasons for it to hold.…”