Brauer–Manin and Descent Obstructions

Abstract

In Chap. 10 we have discussed obstructions to sections arising from arithmetic at p-adic places. Later in Chap. 16 we will discuss what is known about the local analogues of the section conjecture over real and p-adic local fields. The present Chapter concerns the usual next step, when the local problem is considered settled (which it is not for the section conjecture). In order to possibly arise from a common global rational point, the tuple of local solutions must survive known obstructions from arithmetic duality: the Brauer–Manin obstruction and the descent obstruction. We develop here the analogous obstructions to a collection of local sections against being the restriction of a common global section.The Brauer–Manin obstruction for adelic sections was developed in Stix ( J. Pure Appl. Algebra 215(6):1371–1397, 2011). The descent obstruction is actually the older sibling of the Brauer–Manin obstruction. The technique of descent goes back to Fermat. Descent using torsors under tori was developed and studied in detail by Colliot-Thélène and Sansuc, while later Harari and Skorobogatov ( Compos. Math. 130(3):241–273, 2002) analysed descent obstructions coming from non-abelian groups. Descent obstructions under finite groups have been thoroughly analysed by Stoll ( Algebra Number Theor. 1:349–391, 2007). The transfer of the descent obstruction to spaces of sections was essentially worked out in Harari and Stix ( Stix, J. (ed.), Contributions in Mathematical and Computational Science, vol. 2, 2012). Unlike a priori for adelic points, for adelic sections the constant finite descent obstruction turns out to be the only obstruction to globalisation, see Theorem 144. This gives a link between the section conjecture and strong approximation that either yields interesting applications to the section conjecture, see Corollary 158, or, if one is pessimistic, opens up an approach to disproving the section conjecture eventually.