Zassenhaus Filtration and Ihara Constants

Abstract

Let $X$ be a curve over $\mathbb{F}_q$. Denote by $N(X)$ and $g(X)$ the number of $\mathbb{F}_q$-rational points and the genus of $X$ respectively. The Ihara constant \begin{equation*} A(q)=\limsup_{g(X)\to\infty}\frac{N(X)}{g(X)} \end{equation*} measures the asymptotic abundance of rational points on curves over $\mathbb{F}_q$. In this paper, we study the Zassenhaus filtration of pro-$p$ groups arising from class field towers. By combining Golod--Shafarevich theorem, embedding problems and central class field theory, we establish a new criterion for the infinitude of class field towers. As an application, we obtain an improvement on the best known lower bounds of Ihara constant $A(2)$ and $A(3)$ which are explicitly given by \[ A(2)\geq 0.317779\cdots \text{ and } A(3)\geq 0.496074\cdots. \]