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Serre’s modularity conjecture. II. (English) Zbl 1304.11042

The purpose of the article under review is to provide proofs for the main ingredients in the authors’ proof of Serre’s modularity conjecture in Part I [Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041)]. These ingredients are a modularity lifting theorem requiring only weak local assumptions and a theorem embedding a residual representation into a strictly compatible system of Galois representations satisfying strong local requirements.
We now describe both results in more detail. For the precise technical statements the reader is referred to Part I [op. cit.]. For the terminology used here, the review of Part I [op. cit.] can be consulted. Let \(\overline{\rho}\colon\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to \mathrm{GL}_2(\mathbb F)\) with \(\mathbb F/\mathbb F_p\) be an odd absolutely irreducible representation satisfying some rather weak technical conditions. The modularity lifting theorem asserts the following: Let \(\rho\) be any \(p\)-adic lift of \(\overline{\rho}\) that is unramified outside a finite set of primes. Assume that \(\overline{\rho}\) is modular and has a nonsolvable image. If \(p=2\), assume that \(\rho\) is crystalline of weight \(2\) at \(2\), or semistable of weight \(2\) at \(2\) (only when \(k(\overline{\rho}) = 4\)). If \(p>2\), assume that \(\rho\) is crystalline at \(p\) of some weight \(2 \leq k \leq p+1\), or potentially semistable at \(p\) of weight \(2\). Then \(\rho\) is modular. In fact, a more general modularity lifting theorem is proved for a totally real base field (instead of \(\mathbb{Q}\)). The main improvements of this modularity lifting theorem as compared to earlier ones concern particularly the case of residue characteristic \(2\), but also the cases \(k=p\) and \(k=p+1\) for odd \(p\).
The other main result of the article under review provides the existence of a strictly compatible system of Galois representations \((\rho_\iota)\) lifting \(\overline{\rho}\) such that the member at \(p\) is minimally ramified away from \(p\) and away from some special prime \(q\). At \(p\) a simple inertial Weil-Deligne parameter can be prescribed. At the special prime \(q\) extra ramification can be introduced that guarantees a good dihedral behaviour at \(q\) for the reductions of those members of the compatible system that are required in the proof of Serre’s modularity conjecture.
We now mention some of the important ingredients in the proofs of the two theorems. Using Kisin’s framed deformations of Galois representations, the authors prove structural properties (such as relative dimension, regularity, etc.) of local universal deformation rings subject to various deformation conditions. From those, they deduce that the dimensions of the global universal deformation rings with fixed determinant, which they need to consider, are at least one. A patching argument à la Taylor-Wiles, Diamond, Fujiwara and Kisin allows the authors to almost obtain an \(R=\mathbb T\)-theorem. More precisely, they establish a surjection from the global framed deformation ring to the corresponding framed Hecke algebra such that the kernel is of \(p\)-power order, for certain local conditions. The authors moreover provide results that allow one to meet these local conditions after a solvable base change.
The modularity lifting theorem is deduced from this as follows. Since modularity can be tested after solvable base change, the almost \(R=\mathbb T\)-theorem can be applied. It yields that a lift \(\rho\) of \(\overline{\rho}\), which corresponds to a homomorphism from the framed universal deformation ring to the ring of integers in a \(p\)-adic field, factors through the Hecke algebra, thus giving the modularity.
Next we turn to the existence of the compatible system of Galois representations lifting \(\overline{\rho}\). The existence of one lift of \(\overline{\rho}\) subject to the local requirements is implied by the results that the dimension of the global universal deformation ring is at least one and that it is finite as a \(\mathbb Z_p\)-algebra. The latter is a consequence of Taylor’s technique of potential modularity, which is extended in the present article. Finally, the thus produced lift can be made part of a compatible system again by potential modularity and a representation theoretic argument (based on Brauer’s theorem) that had already been used earlier (for instance by L. V. Dieulefait) and that allows one to descend the compatible system arising from potential modularity down to \(\mathbb Q\).

MSC:

11F80 Galois representations
11F11 Holomorphic modular forms of integral weight
11R39 Langlands-Weil conjectures, nonabelian class field theory

Citations:

Zbl 1304.11041
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