First and second $K$-groups of an elliptic curve over a global field of positive characteristic

TL;DR

This paper investigates the structure of K-groups associated with elliptic curves over function fields, demonstrating their divisible properties and explicitly computing related motivic cohomology groups.

Contribution

It establishes the unique divisibility of maximal divisible subgroups of K_1 and K_2 for elliptic curves over function fields and provides explicit calculations of these groups.

Findings

Maximal divisible subgroups of K_1 and K_2 are uniquely divisible.

Explicit computation of K-groups modulo divisible subgroups.

Calculation of motivic cohomology groups of elliptic surfaces.

Abstract

In this paper, we show that the maximal divisible subgroup of groups $K_1$ and $K_2$ of an elliptic curve $E$ over a function field is uniquely divisible. Further those $K$-groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of $E$, which is an elliptic surface over a finite field.