Bounding generators for the kernel and cokernel of the tame symbol for curves

TL;DR

This paper establishes bounds on generators for the kernel and cokernel of the tame symbol on algebraic curves, with potential applications to Chow groups, based on the curve's arithmetic genus.

Contribution

It provides explicit bounds for generators of the tame symbol's kernel and cokernel on curves, a novel result linking algebraic geometry and arithmetic invariants.

Findings

Bounds depend on the arithmetic genus of the curve

Explicit bounds for the cokernel's generators

Potential application to Chow groups

Abstract

Let $C$ be a regular, irreducible curve that is projective over a field. We obtain bounds in terms of the arithmetic genus of $C$ for the generators that are required for the cokernel of the tame symbol, as well as, under a simplifying assumption, its kernel. We briefly discuss a potential application to Chow groups.