On lower bounds for the Ihara constants A(2) and A(3)

TL;DR

This paper improves the known lower bounds for the Ihara constants A(2) and A(3) by applying a variant of Serre's class field tower method to algebraic curves over finite fields.

Contribution

It introduces a modified Serre's class field tower approach to establish better lower bounds for A(2) and A(3).

Findings

Improved lower bounds for A(2) and A(3).

Application of a variant of Serre's method.

Enhanced understanding of rational points on curves.

Abstract

Let X be a curve over the finite field of q elements and let N(X), g(X) be its number of rational points and genus respectively. The Ihara constant A(q) is defined by the limit superior of N(X)/g(X) as the genus of X goes to infinity. 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).