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

✍ Scribed by Stephen A. DiPippo; Everett W. Howe

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 376 KB
• Volume: 73
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2302

✦ Synopsis

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&(1Â2) 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_n matrix [x ei&1 j ], where the e i are positive real numbers.