Real Polynomials with All Roots on the Unit Circle and Abelian Varieties over Finite Fields

AbstractIn this paper we prove several theorems about abelian varieties over finite fields by studying the set of monic real polynomials of degree 2nall of whose roots lie on the unit circle. In particular, we consider a setVnof vectors inRnthat give the coefficients of such polynomials. We calculate the volume ofVnand we find a large easily-described subset ofVn. Using these results, we find an asymptotic formula—with explicit error terms—for the number of isogeny classes ofn-dimensional abelian varieties overFq. We also show that ifn>1, the set of group orders ofn-dimensional abelian varieties overFqcontains every integer in an interval of length roughlyqn−(1/2)centered atqn+1. Our calculation of the volume ofVninvolves the evaluation of the integral over the simplex {(x1,…,xn)| 0⩽x1⩽…⩽xn⩽1} of the determinant of then×nmatrix [xei−1j, where theeiare positive real numbers