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

TL;DR

This paper introduces new upper bounds for the maximum number of rational points on algebraic curves over finite fields, providing exact values for specific cases and exploring related lattice and elliptic curve properties.

Contribution

It presents novel bounds for rational points on curves over finite fields and characterizes Frobenius polynomials for certain genus-12 curves, advancing understanding in algebraic geometry over finite fields.

Findings

N_4(7) = 21

N_8(5) = 29

Frobenius polynomial constraints for genus-12 curves over F_2

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 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 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.