On the K-theory of curves over finite fields

Abstract

Let X be a smooth projective curve over a finite field. The main result is that the odd-dimensional K-theory of the extension of X to the algebraic closure is the sum of two copies of the K-theory of the field. Two plausible conjectures are advanced which would suffice to compute the K-theory of X itself. These provisional computations are then related to the L-functions of X.