More Points Than Expected on Curves over Finite Field Extensions

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