# The conductor density of local function fields with abelian Galois group

## TL;DR

This paper provides an exact formula for counting abelian G-extensions of local function fields with a given conductor bound, and offers a lower bound for the counting problem based on discriminant.

## Contribution

It introduces a precise counting formula for abelian G-extensions of local function fields and establishes a lower bound using discriminant analysis.

## Key findings

- Exact formula for the number of G-extensions up to a conductor bound
- Lower bound for counting G-extensions by discriminant
- Application to local function fields over finite fields

## Abstract

We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.