The Brauer–manin Obstruction for Zero-Cycles on Severi–brauer Fibrations Over Curves

Abstract

Introducing the framework of pseudo-motivic homology, the paper finishes the proof that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on a Severi-Brauer fibration of squarefree index over a smooth projective curve over a number field, provided that the Tate-Shafarevich group of the Jacobian of the base curve is finite. More precisely, for such a variety the Chow group of global zero-cycles is dense in the subgroup of collections of local cycles that are orthogonal to the (cohomological) Brauer group of the variety.

Principe local-global pour les zéro-cycles sur certaines fibrations au-dessus d’une courbe: I

“…D'après le théorème 0.2 (appliqué à id : C → C) de van Hamel [24] (et la remarque 1.1(ii) de [25]), on trouve le diagramme suivant dont la première ligne est une suite exacte.…”

“…Les résultats de ce paragraphe ont été montrés par Colliot-Thélène [3], Frossard [12], et van Hamel [24].…”

Progress concerning the local-global principle for zero-cycles on algebraic varieties

“…A simplied alternative proof was given by Colliot-Thélène [5, Proposition 3.7]; moreover, a statement concerning weak approximation of 0-cycles of degree 1 was also proved in [5, Proposition 3.3]. Summarized by van Hamel, the exactness of (E) for curves was stated in [41].…”

“…The rst dates back to an argument by Colliot-Thélène for 0-cycles on brations over curves of arbitrary genus [6]. Subsequent progress was made by Frossard [18] on brations in Severi-Brauer varieties with the work by van Hamel [41] removing a restriction on the base eld. Finally, in order to remove a subtle restriction on such brations, Wittenberg obtained a general result by reducing the question to brations over the projective line via the Weil restriction with respect to nite morphisms between curves [44].…”

“…Remark 1.1 (ii)]; (ii) the Brauer and Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 [44, Remark 1.1 (iii)]; (iii) the Brauer and Manin obstruction is the only obstruction to weak approximation for 0-cycles of degree 1 [30, Proposition 2.2.1]; (iv) for any integer δ, the Brauer and Manin obstruction is the only obstruction to weak approximation for 0-cycles of degree δ, assuming the existence of a global 0-cycle of degree 1 [30, Proposition 2.2.1].The reformulation above dates back to van Hamel[41].Our main conjecture has been stated by several authors in various forms in the literature, cf., [4, Section 1], [5, Section 1], [11, Section 4], [26, Section 7].…”