Special Number Theory Afternoon

Suppose that $\Sigma \subset \mathbb{C}$ is compact and symmetric about the real axis and is a finite union of rectangles and real intervals with transfinite diameter $d_{\Sigma}>1$. Suppose that $\mu$ is a $H$ older arithmetic probability distribution on $\Sigma$ defined in our work with Orloski with logarithmic energy $I(\mu):=\int\int \log|z_1-z_2|d\mu(z_1)d\mu(z_2)$.

Let $\mathcal{Z}=\{z_1,\dots,z_n\}$ be a generic sampling of $n$ points according to $\mu$. Let $n>m>n^{1-\delta}$ for some $\delta>0$ that depends only on $\mu$. We prove that there are $e^{\frac{n^2}{2}I(\mu)+O(mn+m^2)}$ irreducible polynomial of degree $m+n$ with roots $\alpha_1,\dots,\alpha_{m+n}$ such that $|z_i-\alpha_i|<d_{\Sigma}^{-m}$ for $1\leq i\leq n$ and $\alpha_i \in \Sigma$ for every $1\leq i\leq m+n$.

Given any finite field $\mathbb{F}_q$ 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.