On the Geometry of Tamagawa Numbers of Algebraic Groups

The Algebraic and Anabelian Geometry of Configuration Spaces

In this paper, we study the pro-Σ fundamental groups of configuration spaces, where Σ is either the set of all prime numbers or a set consisting of a single prime number. In particular, we show, via two somewhat distinct approaches, that, in many cases, the “fiber subgroups” of such fundamental groups arising from the various natural projections of a configuration space to lower-dimensional con...

On the Relative Theory of Tamagawa Numbers

This note is an outline of some of the author's recent work on the relative theory of Tamagawa numbers of semisimple algebraic groups. The details and applications will be published elsewhere. Let k be an algebraic number field of finite degree over Q, let G be a connected semisimple algebraic group defined over k. G admits one and only one simply connected covering (G, ir) defined over k (exce...

Tamagawa Numbers of Algebraic Groups Over Function Fields

ec 1 99 7 Tamagawa numbers of polarized algebraic varieties

Let L = (L, ‖ · ‖v) be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety V over a number field F . Denote by N(V,L, B) the number of rational points in V having Lheight ≤ B. In this paper we consider the problem of a geometric and arithmetic interpretation of the asymptotic for N(V,L, B) as B → ∞ in connection with recent conjectures of Fujita concerning the Mini...

Finiteness theorems for algebraic groups over function fields

We prove the finiteness of class numbers and Tate-Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semisimple groups via systematic use of the recently ...