Motivic cohomology and algebraic $K$-theory of some surfaces over finite fields

TL;DR

This paper computes the algebraic K-theory of certain surfaces over finite fields by analyzing motivic cohomology and the motivic Atiyah-Hirzebruch spectral sequence, also extending known cases of Parshin's conjecture.

Contribution

It provides explicit calculations of algebraic K-theory for specific surfaces over finite fields and extends the class of surfaces where Parshin's conjecture holds.

Findings

Computed algebraic K-theory for some surfaces over finite fields.

Calculated motivic cohomology groups relevant to these surfaces.

Extended the class of surfaces satisfying Parshin's conjecture.

Abstract

We compute the algebraic $K$-theory of some classes of surfaces defined over finite fields. We achieve this by first calculating the motivic cohomology groups and then studying the motivic Atiyah-Hirzebruch spectral sequence. In an appendix, we slightly enlarge the class of surfaces for which Parshin's conjecture is known.