# Demostraci\'on simplificada de la Conjetura de Modularidad de Serre (carta)

## TL;DR

This paper simplifies the proof of Serre's Modularity Conjecture over c by leveraging a recent Modularity Lifting Theorem, eliminating complex weight reduction steps used in previous proofs.

## Contribution

It introduces a streamlined proof of Serre's Conjecture that avoids traditional weight reduction techniques by applying Pan's theorem and a new lemma, simplifying the overall argument.

## Key findings

- Proof of Serre's Conjecture over c is simplified
- Weight reduction achieved with fewer applications of Pan's theorem
- Elimination of the need for weight reduction via Galois conjugation

## Abstract

We explain in this letter how using a recent Modularity Lifting Theorem proved by Lue Pan the proofs of Serre's Modularity Conjecture over $\mathbb{Q}$ given by Khare-Wintenberger and the author can be greatly simplified. The main difference with the previous proofs is that neither the process of "weight reduction" developed by Khare in his proof of the level 1 case nor the alternative method of "weight reduction via Galois conjugation" developed by the author are required: weight reduction can now be obtained just with two applications of Pan's result, together with a Lemma that guarantees that the residual representations in the last two steps are not in the bad-dihedral case.