## Serre’s modularity conjecture: the level one case.(English)Zbl 1105.11013

A basic object of arithmetic study is the group $$G_\mathbb Q=\text{Gal}(\overline{\mathbb Q}| \mathbb Q)$$ of all automorphisms of an algebraic closure $$\overline{\mathbb Q}$$ of $$\mathbb Q$$. One may ask for a classification of all representations $$G_{\mathbb Q}\rightarrow\text{GL}_n(\overline{\mathbb F}_l)$$ ($$l$$ prime, $$n>0$$), where $$\overline{\mathbb F}_l$$ is an algebraic closure of the finite field $$\mathbb F_l$$. There is also the corresponding local problem of classifying all representations of the group $$\text{Gal}(\overline{\mathbb Q}_p| \mathbb Q_p)$$ of all continuous automorphisms of an algebraic closure $$\overline{\mathbb Q}_p$$ of $$\mathbb Q_p$$ ($$p$$ prime).
For $$n=1$$, the group $$\text{GL}_1(\overline{\mathbb F}_l)=\mathbb F_l^{\times}$$ is commutative, and the local and the global problems were solved in the first few decades of the last century by erecting the theory of abelian extensions of local and global fields – class field theory. For $$n=2$$, the local problem comes in two flavours, according as $$l=p$$ or $$l\neq p$$, both of which are elementary.
Let us come to the global problem for $$n=2$$. Modular forms give rise (P. Deligne) to representations $$\rho: G_\mathbb Q\rightarrow\text{GL}_2(\overline{\mathbb F}_l)$$ which are odd (i.e. $$\det(\rho(c))=-1$$, for $$c\in G_{\mathbb Q}$$ a complex conjugation), irreducible, and unramified away from finitely many places. In 1972, J.-P. Serre suggested in a letter to H. P. F. Swinnerton-Dyer that all odd, irreducible representations $$\rho:G_{\mathbb Q}\rightarrow\text{GL}_2(\overline{\mathbb F}_l)$$ which are unramified away from finitely many places must arise from some modular form $$f$$ ; this is the qualitative version of Serre’s conjecture.
There is a refined version specifying the weight $$k$$ (resp. the level $$N$$) of $$f$$ in terms of the local behaviour of $$\rho$$ at the prime $$l$$ (resp. at the primes $$p\neq l$$) which made it amenable to computational verification. This version, along with an impressive list of consequences, appears in [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)]. A large number of mathematicians (K. Ribet, B. Mazur, H. Carayol, B. Gross, R. Coleman, J. Voloch, B. Edixhoven, F. Diamond, $$\ldots$$) have contributed to proving that the qualitative version implies the refined version for $$l\neq 2$$.
The level-1 case ($$N=1$$) was mentioned by Serre in a letter to J. Tate in 1973. He replied two months later with a proof for $$l=2$$ [Contemp. Math. 174, 153–156 (1994; Zbl 0814.11057)]; his method, using discriminant bounds, was extended by Serre to $$l=3$$. Here the author proves Serre’s conjecture in the level-1 case for all primes $$l$$ (Theorem 1.1).
As a consequence, one obtains the weight-2 prime-level case for $$l\neq 2$$ (Corollary 1.2) and the finiteness of odd semisimple representations $$G_{\mathbb Q}\rightarrow\text{GL}_2(\overline{\mathbb F}_l)$$ which are unramified outside $$l$$ (Corollary 1.3).
The author builds upon his joint work with J.-P. Wintenberger [arXiv:math/0412076], where (the level-1 case of) the conjecture was proved for $$l=5,7$$.
The proof of Theorem 1.1 may be viewed as taking place in two stages. The first stage consists in producing various $$l$$-adic liftings of $$\rho$$ and in showing that they are part of a compatible family. These liftings are produced using the method of Khare-Wintenberger (loc. cit.), which relies on the potential version of Serre’s conjecture as proved by R. Taylor [J. Inst. Math. Jussieu 1, No. 1, 125–143 (2002; Zbl 1047.11051)] and on a result of G. Böckle on presentations of deformation rings [J. Reine Angew. Math. 509, 199–236 (1999; Zbl 1040.11039)]. The existence of compatible systems is shown using the methods of R. Taylor [Ann. Fac. Sci. Toulouse, VI. Sér., Math. 13, No. 1, 73–119 (2004; Zbl 1074.11030)] and a refinement due to L. Dieulefait [J. Reine Angew. Math. 577, 147–151 (2004; Zbl 1065.11037)].
The second stage consists of a sequence of moves, executed using the first stage, which enables one to prove the level-$$1$$ case of Serre’s conjecture by induction on the prime $$l$$.
Modularity lifting thorems (A. Wiles, R. Taylor, C. Skinner, F. Diamond, K. Fujiwara, M. Kisin) play a crucial role in both the stages.
Let us mention finally that in subsequent joint work of Khare and Wintenberger, building upon the ideas of their previous joint paper and the paper under review, Serre’s conjecture has been proved for $$l\neq 2$$ and $$N$$ odd, and for $$l=2$$ and $$k=2$$. They also reduced the general case to a certain $$2$$-adic modularity lifting result which has very recently been proved by Kisin. All these papers are available on the authors’ websites. Thus, finally, there is now a complete proof of Serre’s conjecture for all levels.

### MSC:

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

### Keywords:

Serre’s modularity conjecture
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### References:

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