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Abstract: A fundamental problem in the theory of curves over finite fields is to determine the sets there is a curve over of genus g with exactly N rational points}. A complete description of is out of reach. So far, mostly bounds for the numbers have been studied. In particular, Elkies et al. proved that there is a constant , such that for any there is some with . This implies that , and solves a long-standing problem by Serre. We extend the result of Elkies et al. substantially: there are constants such that for all , the whole interval is contained in . [Copyright &y& Elsevier]