The similarity between finite-dimensional projective spaces and finite abelian groups has often been noted(1); and thus one may expect that the more general features of these two theories are identical. But the likeness is more than a superficial one; and consequently it is possible to give a unified treatment for spaces and groups. It may be worthwhile to indicate in a few lines the developmen...

LetX be a smooth projective geometrically connected curve over a finite field with function field K. Let G be a connected semisimple group scheme over X . Under certain hypothesis we prove the equality of two numbers associated with G. The first is an arithmetic invariant, its Tamagawa number. The second, is a geometric invariant, the number of connected components of the moduli stack of G-tors...

Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. In the case of R containing a field of characteristic zero we show that if (V, φ)⊗R K is isotropic over K, then (V, φ) is isotropic over R. 1 Characteristic zero case 1.0.1 Theorem (Main). Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. Suppose R contains a f...

The problem of calculating the trace of an elliptic curve over a finite field has attracted considerable interest in recent years. There are many good reasons for this. The question is intrinsically compelling, being the first nontrivial case of the natural problem of counting points on a complete projective variety over a finite field, and figures in a variety of contexts, from primality provi...

We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes with the homotopy type of an Eilenberg-MacLane space K(π, 1) equipped with a cellular self-map of prime order. As an application we derive the section conjecture for the real points of aspherical varieties defined over the field of real numbers and the natural an...