[Paper Review] How to facet a gemstone: from potential modularity to the proof of Serre's modularity conjecture

Luís Dieulefait|ArXiv.org|Dec 9, 2007

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper presents a comprehensive survey of the proof of Serre’s modularity conjecture, leveraging potential modularity theorems (notably Taylor’s) and modularity lifting techniques to propagate modularity from base cases. By using tools like minimal lifts, weight reduction via Galois conjugates, and iterated ramification killing, the authors reduce the general case to level 1 and small weights, ultimately proving modularity for all odd conductor, odd, irreducible 2-dimensional Galois representations.

ABSTRACT

In this survey paper we present recent results obtained by Khare, Wintenberger and the author that have led to a proof of Serre's conjecture, such as existence of compatible families, modular upper bounds for universal deformation rings and existence of minimal lifts, prime switching and modularity propagation, weight reduction (via existence of conjugates) and (iterated) killing ramification. The main tools used in the proof of these results are modularity lifting theorems a la Wiles and a result of potential modularity due to R. Taylor.

Motivation & Objective

• To explain the key steps and tools used in the proof of Serre’s modularity conjecture for odd, irreducible 2-dimensional Galois representations.
• To clarify how potential modularity and modularity lifting theorems enable the propagation of modularity from simple base cases to general cases.
• To detail the role of minimal lifts, weight reduction via Galois conjugates, and iterated ramification killing in reducing the problem to manageable cases.
• To demonstrate how the introduction of a good-dihedral prime in the level facilitates the reduction process and ensures large residual images.
• To show how the level 1, weight 2 case and its generalizations are reduced to known modular cases using characteristic switching and semistable reduction properties.

Proposed method

• Utilize R. Taylor’s potential modularity theorem to establish the existence of compatible families of Galois representations.
• Apply modularity lifting theorems à la Wiles and Kisin to propagate modularity from residual representations to lifts.
• Implement 'prime switching' to move between different characteristics, simplifying the representation at each step.
• Use 'weight reduction' via Galois conjugates to connect representations of weight $k$ to those of weight approximately $k/2$, reducing the problem to smaller weights.
• Perform 'iterated killing of ramification' by successively removing primes from the level, using semistable lifts and characteristic switching to satisfy technical conditions.
• Introduce a good-dihedral prime in the level to ensure large residual images, enabling the application of modularity lifting theorems in characteristic 2 and beyond.

Experimental results

Research questions

• RQ1 How can potential modularity theorems be combined with modularity lifting theorems to prove Serre’s conjecture?
• RQ2 What role does the existence of minimal lifts play in reducing the level and weight of Galois representations?
• RQ3 How does the introduction of a good-dihedral prime in the level facilitate the proof of modularity for odd conductor representations?
• RQ4 In what way does the 'existence of Galois conjugates' enable weight reduction from $k$ to approximately $k/2$?
• RQ5 How can iterated ramification killing be used to reduce the level to 1, and what technical conditions are required for this process?

Key findings

• The proof of Serre’s modularity conjecture for odd, irreducible 2-dimensional Galois representations is achieved by reducing the problem to the level 1, weight 2 case via a sequence of technical moves.
• The level 1, weight 2 case is established using modularity lifting theorems and the fact that such representations are known to be modular since 1973.
• Weight reduction via Galois conjugates allows the reduction of the general weight $k$ case to weight $k/2$, ultimately connecting to the weight 2 case.
• The introduction of a good-dihedral prime ensures that the residual image is large, which is essential for applying modularity lifting theorems in characteristic 2.
• Iterated ramification killing, combined with characteristic switching and semistable lifts, reduces the level to 3, and further reductions via Sophie Germain primes lead to the level 21, weight 2 case.
• The final reduction to level 3 and weights 4 and 6 is completed by switching to characteristic 7 and using the fact that semistable abelian varieties of small conductor are modular, as shown by Schoof.