Curves over higher local fields

TL;DR

This paper proves the vanishing of certain cohomological groups over higher local fields, generalizing known properties of finite and one-dimensional local fields, and applies this to study curves over such fields.

Contribution

It establishes a new cohomological vanishing result for higher local fields and explores its implications for the arithmetic of curves defined over these fields.

Findings

Proves vanishing of two cohomological groups over higher local fields

Generalizes properties from finite and one-dimensional local fields

Provides applications to the arithmetic of curves over higher local fields

Abstract

In this work, we prove the vanishing of the two cohomological group of the higher local field. This generalize the well-known propriety of finite field and one dimensional local field. We apply this result to study the arithmetic of curve defined over higher local field.