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 M q ( g ) : = { N ∈ N | there is a curve over F q of genus g with exactly N rational points}. A complete description of M q ( g ) is out of reach. So far, mostly bounds for the numbers N q ( g ) : = max M q ( g ) have been studied. In particular, Elkies et al. proved that there is a constant γ q > 0 , such that for any g ⩾ 0 there is some N ∈ M q ( g ) with N ⩾ γ q g . This implies that lim inf g → ∞ N q ( g ) / g > 0 , and solves a long-standing problem by Serre. end the result of Elkies et al. substantially: there are constants α q , β q > 0 such that for all g ⩾ 0 , the whole interval [ 0 , α q g − β q ] ∩ N is contained in M q ( g ) .