Counting Points on Curves over Finite Fields

We consider the problem of counting the number of points on a plane curve, defined by a homogeneous polynomialF(x,y,z) q[x,y,z], which are rational over a ground field q. More precisely, we show that if we are given a projective plane curve of degreen, and if has only ordinary multiple points, then one can compute the number of q-rational points on in randomized time (logq)Δwhere Δ = nO(1). Since our algorithm actually computes the characteristic polynomial of the Frobenius endomorphism on the Jacobianof , it ,follows that we may also compute (1) the number of q-rational points on the smooth projective model of , (2) the number of q-rational points on the Jacobian of , and (3) the number of qm-rational points on in any given finite extension qmof the ground field, each in a similar time bound.