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

Abstract

We provide new upper bounds on $N_q(g)$, the maximum number of rational points on a smooth absolutely irreducible genus-$g$ curve over ${\mathbb{F}}_q$, for many values of $q$ and $g$. Among other results, we find that $N_4(7)=21$ and $N_8(5)=29$, and we show that a genus-12 curve over ${\mathbb{F}}_2$ having 15 rational points must have characteristic polynomial of Frobenius equal to one of three explicitly given possibilities.

We also provide sharp upper bounds for the lengths of the shortest vectors in Hermitian lattices of small rank and determinant over the maximal orders of small imaginary quadratic fields of class number 1.

Some of our intermediate results can be interpreted in terms of Mordell–Weil lattices of constant elliptic curves over one-dimensional function fields over finite fields. Using the Birch and Swinnerton-Dyer conjecture for such elliptic curves, we deduce lower bounds on the orders of certain Shafarevich–Tate groups.