(An official version and an arXiv preprint are available.)

As usual, let *N _{q}*(

As before, our methods involve analyzing the possible Weil polynomials for
a curve over **F**_{q} with a given number *N* of points. Given a Weil polynomial
of an isogeny class of abelian variety over **F**_{q}, we try to find reasons why
the isogeny class cannot contain a Jacobian — or, if we cannot find a
reason to exclude a Jacobian, we try to prove that a curve whose Jacobian
lies in the given isogeny class must have special properties that allow us
to efficiently search for it.

There are four main new techniques presented in this paper. First, we
show that in many of the arguments in our previous paper where we made use
of the resultant of two factors of a possible real Weil polynomial, we can
instead use the “reduced resultant.” Second, we generalize an argument
from our earlier paper that says that when *q = s ^{2}*, there is no curve
whose real Weil polynomial is (

Van der Geer and van der Vlugt have maintained tables of the best known
lower and upper bounds on *N _{q}*(

The van der Geer / van der Vlugt tables have been incorporated into an
online database,
www.manypoints.org,
that includes the best known bounds
on *N _{q}*(

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.

Here are links to some of the Magma programs related to this paper:

- Most of our upper bounds on
*N*(_{q}*g*) come directly from the routines in IsogenyClasses.magma. Results from running this program were used to populate many entries in the tables at manypoints.org. - To show that
*N*_{32}(4) ≤ 72 we had to enumerate the genus-4 double covers of the elliptic curve*y*over^{2}+ xy = x^{3}+ x**F**_{32}and show that none of them has more than 72 points. Code for doing this can be found here. This code also produces examples of genus-4 curves over**F**_{32}with 71 points. - For some of our improvements we used sharp upper bounds on the lengths of short vectors in Hermitian lattices over the rings of integers of imagainary quadratic fields of class number 1. We computed these bounds using the programs in HermitianForms.magma.