Higher K-groups of smooth projective curves over finite fields

Let X be a smooth projective curve over a finite field F with q elements. For m⩾1, let Xm be the curve X over the finite field Fm, the m-th extension of F. Let Kn(m) be the K-group Kn(Xm) of the smooth projective curve Xm. In this paper, we study the structure of the groups Kn(m). If l is a prime, we establish an analogue of Iwasawa theorem in algebraic number theory for the orders of the l-primary part Kn(lm){l} of Kn(lm). In particular, when X is an elliptic curve E defined over F, our method determines the structure of Kn(E). Our results can be applied to construct an efficient DL system in elliptic cryptography.