On the Zeta Functions of an optimal tower of function fields over $\FF_4$

TL;DR

This paper derives explicit formulas for the zeta functions of the first six function fields in the second Garcia-Stichtenoth tower over a74, using Jacobian decomposition techniques, advancing understanding of their arithmetic properties.

Contribution

It introduces a recursion for the zeta functions of the tower's function fields over a74, enabling explicit computation and analysis.

Findings

Explicit zeta functions for the first six fields obtained

Recursion formula derived for the zeta functions

Application of Jacobian decomposition to tower analysis

Abstract

In this paper we derive a recursion for the zeta function of each function field in the second Garcia-Stichtenoth tower when $q=2$. We obtain our recursion by applying a theorem of Kani and Rosen that gives information about the decomposition of the Jacobians. This enables us to compute the zeta functions explicitly of the first six function fields.