On the number of rational points of curves over finite fields

A fundamental problem in the theory of curves over finite fields is to determine the sets Mq(g):={N∈N| there is a curve over Fq of genus g with exactly N rational points}. A complete description of Mq(g) is out of reach. So far, mostly bounds for the numbers Nq(g):=maxMq(g) have been studied. In particular, Elkies et al. proved that there is a constant γq>0, such that for any g⩾0 there is some N∈Mq(g) with N⩾γqg. This implies that , and solves a long-standing problem by Serre.We extend the result of Elkies et al. substantially: there are constants αq,βq>0 such that for all g⩾0, the whole interval [0,αqg−βq]∩N is contained in Mq(g).