Dang-Khoa Nguyen: The Pólya-Carlson dichotomy for some dynamical zeta functions (NTWS 215)

Let $\theta$ be a map from a set $X$ to itself. Suppose that for $k\geq 1$, the number $N_k(\theta)$ of fixed points of the $k$-th fold iterate $\theta^k=\theta\circ\cdots\circ\theta$ is finite. Then we can define the dynamical or Artin-Mazur zeta function

$$\zeta_{\theta}(z)=\exp\left(\sum_{k=1}^{\infty}\frac{N_k(\theta)}{k}z^k\right).$$

A complex power series with radius of convergence $R\in (0,\infty)$ is said to satisfy the P\'olya-Carlson dichotomy if it is either a rational function or it cannot be extended analytically beyond the disk of radius $R$.
In this talk, we discuss the Pólya-Carlson dichotomy for the Artin-Mazur zeta functions of endomorphisms of tori and abelian varieties. This is from a joint work with Bell, Gunn, and Saunders and another with Baril Boudreau and Holmes.

Link to slides: https://drive.google.com/file/d/1I9VD...

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date:
Thursday, June 6, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 7, 2024 (1am AEST, 3am NZST)