Number of points of certain Artin-Schreier curves

TL;DR

This paper proves a conjecture linking exponential sums and the point counts of specific Artin-Schreier curves over finite fields, impacting the understanding of their L-polynomials.

Contribution

It establishes the equality of point counts for certain Artin-Schreier curves, confirming a conjecture and revealing new insights into their L-polynomials.

Findings

Artin-Schreier curves have the same number of points over finite fields

Confirmed a conjecture relating exponential sums and curve point counts

Implications for the L-polynomials of these curves

Abstract

We prove a conjecture of Johansen, Helleseth and Kholosha concerning equality of exponential sums related to the cross-correlation of $m$-sequences. In the proof we show that certain Artin-Schreier curves have the same number of points over finite fields. This has a consequence regarding the L-polynomials of these curves.