(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 2n all of whose roots lie on the unit circle. In particular, we consider a set V_n of vectors in Rⁿ 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 qⁿ+1. Our calculation of the volume of V_n involves the evaluation of the integral over the simplex {(x₁,...,x_n) | 0 < x₁ < ... < x_n < 1 } of the determinant of the n by n matrix whose (i,j)th entry is x_j^e_i-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.