Stephen A. DiPippo and Everett W. Howe: Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory 73 (1998) 426–450, MR 2000c:11101, Zbl 0931.11023. Corrigendum, J. Number Theory 83 (2000) 182.

(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 Vn of vectors in Rn that give the coefficients of such polynomials. We calculate the volume of Vn and we find a large easily-described subset of Vn. Using these results, we find an asymptotic formula — with explicit error terms — for the number of isogeny classes of n-dimensional abelian varieties over Fq. We also show that if n>1, the set of group orders of n-dimensional abelian varieties over Fq contains every integer in an interval of length roughly qn-1/2 centered at qn+1. Our calculation of the volume of Vn involves the evaluation of the integral over the simplex {(x1,...,xn) | 0 < x1 < ... < xn < 1 } of the determinant of the n by n matrix whose (i,j)th entry is xjei-1, where the ei are positive real numbers.

NOTE: There is a misprint in the published version of the paper. The expression qn(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.