Poincar� polynomials of the variety of stable bundles

Abstract

§1. In this note, we indicate a few improvements to [3]. Let X be an irreducible, non-singular projective algebraic curve defined over a finite field Fq with q elements, of characteristic p. Let SL(n, d) be the coarse moduli scheme of isomorphism classes of stable vector bundles of rank n and determinant isomorphic to an lFq-rational line bundle L of degree d. [We will assume that (n, d) = 1.] By replacing lFq by a finite extension if necessary, we may assume that SL(n, d) is defined over IFq. As one might expect, it is then indeed true that the IFq-rational points of SL(n, d) are precisely the stable vector bundles on X defined over IFq (see [3]). By the Weil conjectures, it is easy to write down the Poincar6 polynomials of SL(n, d) once the number of its lFq-rational points is known. In order to compute the latter, one first notices that the fact that the Tamagawa number of SL(n) is 1 can be interpreted as follows.