Counting Irreducible Integral Polynomials with Roots Approximating Configuration of Points

Suppose that Σ⊂ℂ is compact and symmetric about the real axis and is a finite union of rectangles and real intervals with transfinite diameter dΣ greater than 1. Suppose that μ is a H older arithmetic probability distribution on Σ defined in our work with Orloski with logarithmic energy I(μ):=∫∫log|z1−z2|dμ(z1)dμ(z2). Let ={z1,…,zn}  be a generic sampling of n points according to μ. Let n greater than m greater than n1−δ for some δ greater than 0 that depends only on μ. We prove that there are en22I(μ)+O(mn+m2) irreducible polynomial of degree m+n with roots α1,…,αm+n such that |zi−αi| greater than d−mΣ for 1 less than or equal to i less than or equal to n and αi∈Σ for every 1 less than or equal to i less than or equal to m+n. Given any finite field 𝔽q and any integer n, we give an asymptotic formula for the number of isogeny classes of ordinary Abelian varieties of dimension n with prescribed Frobenius eigenvalue angles in small intervals.