(An official electronic version and an official correction are available. A corrected unofficial electronic version is also available.)

In this paper we prove several theorems about abelian varieties
over finite fields by studying the set of monic real polynomials
of degree 2*n* all of whose roots lie on the unit circle.
In particular, we consider a set *V*_{n} of vectors in
**R**^{n} that give the coefficients of such polynomials.
We calculate the volume of *V*_{n} and we find a large
easily-described subset of *V*_{n}. Using these results,
we find an asymptotic formula — with explicit error terms — for
the number of isogeny classes of *n*-dimensional abelian
varieties over **F**_{q}. We also show that if *n*>1,
the set of group orders of *n*-dimensional abelian varieties over
**F**_{q} contains every integer in an interval of
length roughly *q*^{n-1/2} centered at *q*^{n}+1.
Our calculation of the volume of *V*_{n} involves the
evaluation of the integral over the simplex
{(*x*_{1},...,*x*_{n}) |
0 < *x*_{1} < ... < *x*_{n} < 1 }
of the determinant of the *n* by *n* matrix whose (*i,j*)th entry is
*x*_{j}^{ei-1},
where the *e*_{i} are positive real numbers.

** NOTE: There is a misprint in the published version of the
paper.**
The expression *q*^{n(n-1)/4}
should be replaced with *q*^{(n+2)(n-1)/4}
in the first displayed equation of Theorem 1.2, in the first
displayed equation of Proposition 3.2.1, and in the displayed
equation at the bottom of page 445. Thanks go to
Joshua Holden
for pointing this out to us.