Meeting Details

Title: Statistics for zeros of zeta functions in a family of curves Algebra and Number Theory Seminar Maosheng Xiong, Penn State University Let $\mathbb{F}_q$ be a finite field of cardinality $q$ and $l \ge 2$ be a positive integer such that $q \equiv 1 \pmod{l}$. Extending the work of Faifman and Rudnick on hyperelliptic curves, we study the distribution of zeros of zeta functions of a family of curves over $\mathbb{F}_q$ having a cyclic $l$-to-1 map to $\mathbb{P}^1(\mathbb{F}_q)$, in the limit of large genus. The zeros all lie on a circle, according to the Riemann Hypothesis for curves, and their angles are uniformly distributed, so for a curve of genus $g$ a fixed interval $I$ contains asymptotically $2g|I|$ angles as the genus grows. We show that the variance of number of angles in $I$ is asymptotically $(l-1)\frac{2}{\pi^2} \log (2g |I|)$ and the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles $2g|I|$ tends to infinity.