The level 1 case of Serre's conjecture revisited

TL;DR

This paper provides a new proof of the level 1 case of Serre's conjecture by establishing the existence of conjugate Galois representations and introducing a weight reduction method.

Contribution

It introduces a novel approach using conjugate Galois representations to simplify the proof of the level 1 case of Serre's conjecture.

Findings

Existence of conjugate Galois representations proven

A simple weight reduction method developed

Alternative proof of the level 1 case of Serre's conjecture provided

Abstract

We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre's conjecture follows.