 
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 ( : SK1(X) K1(k)) and the onedimensional cycle group W(X) = Coker (
:
K2(k) H0
Zar(X, K2)), 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
