On the section conjecture and Brauer-Severi varieties

Abstract

J. Stix proved that a curve of positive genus over Q which maps to a non-trivial Brauer-Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction R k/Q X admits a rational map to a non-trivial Brauer-Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer-Severi variety P such that the corestriction cor k/Q ([P]) ∈ Br(Q) is non-trivial, then X satisfies the section conjecture.Let X be a geometrically connected variety over a field k with separable closure k, there is a short exact sequence of étale fundamental groupsGrothendieck's section conjecture predicts that, if X is a smooth, projective curve of genus at least 2 and k is a number field, the set of rational points X(k) is in natural bijection with sections of the sequence above modulo the action of π 1 (X¯k) by conjugation.Thanks to an idea of Tamagawa [Tam97] [Sti13, Corollary 102] it is sufficient to prove the conjecture for curves with no rational points, and some results have been proved about such curves. The section conjecture holds for X if• the number field k has a real place k → R such that X R (R) = ∅, see [Moc03, Corollary 3.13], or • the class of Pic 1X in H 1 (k, Pic 0 X ) is not divisible, see [HS09, Theorem 1.2], or • k = Q and X maps to a non-trivial Brauer-Severi variety, see [Sti10a, Corollary 18]. The last condition, which is due to Stix, holds over any number field k, but with an additional hypothesis: for every prime number p, it is required that X has bad reduction at most at one place p of k over p, see [Sti10a, Theorem 17]. We provide a different generalization based on Weil's restriction of scalars.Recall that, given a finite separable extension k/h of fields and a quasi-projective variety X over k, the Weil restriction R k/h X is a quasi-projective variety over h characterized by a functorial bijection Hom(S, R k/h X) Hom(S k , X) for schemes S over h. In particular, k-rational points of X are in natural bijection with h-rational points of R k/h X.If X is a curve of genus g over a number field k, then (R k/Q X) Q is a product of [k : Q] curves of genus g, so passing to the Weil restriction is basically a trade-off between the complexity of the base field and the complexity of the variety. We prove the following.Theorem 1. Let X be a smooth, projective, geometrically connected curve of positive genus over a number field k. Assume that R k/Q X admits a rational map to a non-trivial Brauer-Severi variety. Then the The author is supported by the DFG Priority Program "Homotopy Theory and Algebraic Geometry" SPP 1786.