A calculation of $L$-series in terms of Jacobi sums

TL;DR

This paper derives a method to compute the coefficients of Dirichlet L-series for cyclic function field extensions using Jacobi sums, with applications demonstrated in the final section.

Contribution

It introduces a novel calculation technique expressing L-series coefficients as linear combinations of Jacobi sums for cyclic extensions.

Findings

Coefficients of L-series are expressed via Jacobi sums.

Method applicable to cyclic extensions over finite fields.

Applications of the calculation are demonstrated.

Abstract

Let us consider a cyclic extension of a function field defined over a finite field. For a character (non-trivial) of this extension, we calculate, as a linear combinations of products of Jacobi sums, the coefficients of the polynomial given by its Dirichtlet $L$-series. In the last section we show applications of this calculation.