Euler-Poincare Characteristics for Motivic Cohomology and Arithmetic Groups

Abstract

Euler-Poincare characteristics have a way of cropping up when one studies the values of zeta functions at integers. On the one hand, they arise in arithmetic versions of the Gauss-Bonnet theorem [On], [H], [S], [T2], and, on the other, in applications of etale cohomology and of Ktheory to varieties over finite fields [L1-4], [BN], [Sch], [M2-3]. Here we investigate some consequences of the Lichtenbaum-Milne theory and of Serre's theory of cohomology of S-arithmetic groups as they apply to elliptic surfaces over finite fields. Some of Serre's definitions are recalled briefly at the beginning of Section 1. This introduction is followed by a calculation of the EulerPoincare characteristic of a v-arithmetic of units in a maximal order of a algebra over a global field of positive characteristic. In Section 2 we show how to associate such a quaternion group to each irreducible fiber of an elliptic surface defined over a finite field and carrying a level two structure. The question then is how the Euler characteristics of these groups relate to the arithmetic of the surface. Their product diverges, but if each Euler characteristic is appropriately weighted, the product converges to the motivic Euler characteristic of the constant sheaf Z in the etale topology. (The statement of the theorem is not quite this clean as it is necessarily smudged by the presence of some global constants and by factors coming from the singular fibers. Two remarks following the theorem indicate how it might be polished.) The paper closes with an appendix that studies in detail the association of an algebraic family of algebras to the smooth fibers of an elliptic surface with level two structure. This construction may be of independent interest for the study of the class field theory of the surface.