 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000000
S 00029947(XX)00000
CYCLES ON CURVES OVER GLOBAL FIELDS OF POSITIVE
CHARACTERISTIC
REZA AKHTAR
Abstract. Let k be a global field of positive characteristic and # : X # Spec k
a smooth projective curve. We study the zerodimensional cycle group V (X) =
Ker (# # : SK 1 (X) # K 1 (k)) and the onedimensional cycle group W (X) = Coker (# # :
K 2 (k) # H 0
Zar (X, K 2 )), addressing the conjecture that V (X) is torsion and W (X)
is finitely generated. The main idea is to use Abhyankar's Theorem on resolution
of singularities to relate the study of these cycle groups to that of the Kgroups of
a certain smooth projective surface over a finite field.
1. Introduction
Let k be a global field of positive characteristic; that is, a field which is finitely
generated and of transcendence degree one over a finite field. Let # : X # Spec k
be a smooth projective curve over k; consider the cycle groups
V (X) = Ker (# # : SK 1 (X) # K 1 (k)
