  We resolve a 1983 question of Serre by constructing curves with many points
of every genus over every finite field. More precisely, we show that for every
prime power q there is a positive constant c_q with the following property: for
every non-negative integer g, there is a genus-g curve over F_q with at least
c_q * g rational points over F_q. Moreover, we show that there exists a
positive constant d such that for every q we can choose c_q = d * (log q). We
show also that there is a constant c > 0 such that for every q and every n > 0,
and for every sufficiently large g, there is a genus-g curve over F_q that has
at least c*g/n rational points and whose Jacobian contains a subgroup of
rational points isomorphic to (Z/nZ)^r for some r > c*g/n.
