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We provide new upper bounds on N_q(g), the maximum number of rational pointson a smooth absolutely irreducible genus-g curve over F_q, for many values of qand g. Among other results, we find that N_4(7) = 21 and N_8(5) = 29, and weshow that a genus-12 curve over F_2 having 15 rational points must havecharacteristic polynomial of Frobenius equal to one of three explicitly givenpossibilities.  We also provide sharp upper bounds for the lengths of the shortest vectors inHermitian lattices of small rank and determinant over the maximal orders ofsmall imaginary quadratic fields of class number 1.  Some of our intermediate results can be interpreted in terms of Mordell-Weillattices of constant elliptic curves over one-dimensional function fields overfinite fields. Using the Birch and Swinnerton-Dyer conjecture for such ellipticcurves, we deduce lower bounds on the orders of certain Shafarevich-Tategroups.

New methods for bounding the number of points on curves over finite fields