On lower bounds for the Ihara constants and

Abstract

AbstractLet $ \mathcal{X} $ be a curve over ${ \mathbb{F} }_{q} $ and let $N( \mathcal{X} )$, $g( \mathcal{X} )$ be its number of rational points and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)= {\mathrm{lim~sup} }_{g( \mathcal{X} )\rightarrow \infty } N( \mathcal{X} )/ g( \mathcal{X} )$. In this paper, we employ a variant of Serre’s class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$.