Real polynomials with all roots on the unit circle and abelian varieties over finite fields, by Stephen A. DiPippo and Everett W. Howe

(This paper is an improved version of the paper of the same title that appeared in these archives in July 1997. We've sharpened the results, added an explicit error bound to the main theorem, and figured out how to take account of the non-ordinary varieties.)

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.

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



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