Real polynomials with all roots on the unit circle and abelian varieties over finite fields

Abstract

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^{e_i-1}, where the e_i are positive real numbers.