Cohomological Hasse principle and motivic cohomology for arithmetic schemes

In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.Comment: 47 pages, final versio

Concentration or representation : the struggle for popular sovereignty

There is a tension in the notion of popular sovereignty, and the notion of democracy associated with it, that is both older than our terms for these notions themselves and more fundamental than the apparently consensual way we tend to use them today. After a review of the competing conceptions of 'the people' that underlie two very different understandings of democracy, this article will defend what might be called a 'neo-Jacobin' commitment to popular sovereignty, understood as the formulation and imposition of a shared political will. A people's egalitarian capacity to concentrate both its collective intelligence and force, from this perspective, takes priority over concerns about how best to represent the full variety of positions and interests that differentiate and divide a community

Campana points of bounded height on vector group compactifications

We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behaviour of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our set-up, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups.Comment: 52 pages; minor revision, changes in the definition of Campana point