In this paper we prove several theorems about abelian varieties
over finite fields by studying the set of monic real polynomials
of degree 2n 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.

This paper has appeared: J. Number Theory 73 (1998) 426-450.

- Isogeny.dvi (104000 bytes) [1998 Mar 23]
- Isogeny.dvi.gz (41406 bytes)
- Isogeny.ps.gz (111207 bytes)

Stephen A. DiPippo and Everett W. Howe <however@alumni.caltech.edu>