On the coefficients of the Zeta-function's L-polynomial for algebraic function fields over finite constant fields

TL;DR

This paper derives explicit formulas for the coefficients of the Zeta-Function's L-polynomial in algebraic function fields over finite fields, enabling computation of class numbers and applications to specific curve cases.

Contribution

It provides a new explicit formula for the coefficients of the Zeta-Function's L-polynomial in algebraic function fields over finite fields, linking it to class number calculations.

Findings

Explicit coefficient formulas for Zeta-Function's L-polynomial

Expression of class numbers for algebraic function fields

Application to curves of defect 2 over $\\mathbb{F}_2$

Abstract

We give an explicit formula of the coefficients of the Zeta-Function's L-polynomial for algebraic function fields over finite constant fields. Thus, we deduce an expression of the class number of algebraic function fields defined over finite fields. Moreover, we give an application of this formula in the case of the curves of defect 2 defined over $\mathbb{F}_2$.