More points than expected on curves over finite field extensions

On average, there are q(r) + o(q(r/2)) F-q-rational points on curves of genus g defined over F-qr. This is also true if we restrict our average to genus g curves defined over F-q, provided r is odd or r > 2g. However, if r = 2,4,6,... or 2g then the average is q(r) + q(r/2) + o(q(r/2)). We give a number of proofs of the existence of these q(r/2) extra points, and in some cases give a precise formula, but we are unable to provide a satisfactory explanation for this phenomenon. (C) 2000 Academic Press.