The authors’ article under review, together with Part II [Invent. Math. 178, No. 3, 505–586 (2009; Zbl 1304.11042)] and M. Kisin [Invent. Math. 178, No. 3, 587–634 (2009; Zbl 1304.11043)], proves Serre’s modularity conjecture [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)]. In the reviewer’s opinion, this is one of the major achievements in arithmetic geometry so far. It relies on deep work of, besides the authors, Böckle, Breuil, Buzzard, Carayol, Coleman, Conrad, Deligne, Diamond, Dieulefait, Edixhoven, Fontaine, Fujiwara, Gross, Hida, Kisin, Langlands, Mazur, Ramakrishna, Ribet, Saito, Serre, Shimura, Skinner, Taylor, Wiles, and many others.
Serre’s modularity conjecture proposes a complete and very explicit classification of all odd irreducible two-dimensional mod \(p\) representations of \(\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\) in terms of modular forms. Among its important consequences are the generalized Taniyama-Shimura-Weil conjecture for abelian varieties of \(\mathrm{GL}_2\)-type over \(\mathbb Q\) and Artin’s conjecture for odd two-dimensional complex representations of \(\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\).
In this review, we first explain Serre’s modularity conjecture and its consequences in some detail and then give an overview of the proof.
Let \(f\) be a cuspidal modular form which is an eigenfunction for all Hecke operators; assume that \(f\) is given by a Fourier series of the form \(q+\sum_{n=2}^\infty a_n q^n\) with \(q = q(z) = e^{2\pi iz}\) and that \(f\) has level \(N\), weight \(k\) and nebentype character \(\chi\). For any prime \(p\) and any embedding \(\iota\colon \mathbb Q(a_n;\, n \in\mathbb N) \hookrightarrow \overline{\mathbb Q}_p\), by a theorem of Shimura, Deligne and Deligne-Serre there is an odd (i.e. the determinant of the image of any complex conjugation is \(-1\)) irreducible Galois representation \(\rho_{f,\iota}\colon\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to\mathrm{GL}_2(\overline{\mathbb Q}_p)\) such that the characteristic polynomial at any prime \(q \nmid Np\) is equal to \(X^2 - \iota(a_q) X + \iota(\chi(q) q^{k-1})\). By choosing an integral model for \(\rho_{f,\iota}\), reducing it modulo the maximal ideal and passing to the semi-simplification, one obtains a continuous Galois representation \(\overline{\rho}_{f,\iota}\colon \mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to \mathrm{GL}_2(\overline{\mathbb F}_p)\). This representation, however, need not be irreducible. Representations arising as \(\rho_{f,\iota}\) or \(\overline{\rho}_{f,\iota}\) are called modular. A continuous Galois representation \(\overline{\rho}\colon\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to \mathrm{GL}_2(\overline{\mathbb F}_p)\) is said to be of \(S\)-type if it is odd and irreducible.
Serre’s modularity conjecture (in its qualitative form) and hence the main result of this article and its sequel, when combined with [M. Kisin, loc. cit.], is the theorem: Every \(\overline{\rho}\) of \(S\)-type is modular.
The refined or quantitative form of Serre’s modularity conjecture also specifies a weight \(k(\overline{\rho})\) and a level \(N(\overline{\rho})\) for a newform giving rise to \(\overline{\rho}\) according to the principle that the ramification of \(\overline{\rho}\) away from \(p\) is taken account of by the level (\(N(\overline{\rho})\) is taken to be the Artin conductor of \(\overline{\rho}\) outside of \(p\)), whereas the weight \(k(\overline{\rho})\) contains information on the ramification at \(p\). The equivalence of the qualitative and the quantitative form of the conjecture had been established earlier.
For a number field \(E\), an \(E\)-rational compatible system of \(2\)-dimensional Galois representations is (roughly speaking) a family \((\rho_\iota)\), indexed by the embeddings \(\iota\colon E \to \overline{\mathbb Q}_p\) for all primes \(p\), where the \(\rho_\iota\colon \mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to \mathrm{GL}_2(\overline{\mathbb Q}_p)\) are continuous semisimple geometric representations such that the Weil-Deligne representations at all primes and the Hodge-Tate weights \((a,b)\) are independent of \(\iota\). It is called regular if \(a \neq b\) and irregular otherwise. A newform \(f\) of weight \(k\) gives rise to a compatible system of Galois representations (by the theorem recalled above), which is regular if and only if \(k \geq 2\). A consequence of the main theorem of the present article is the modularity of compatible systems of Galois representations: Any two-dimensional compatible system of Galois representations over \(\mathbb{Q}\) which is odd and irreducible (meaning that all members satisfy these conditions) arises from a newform (up to twist).
We recall that a simple abelian variety \(A\) over \(\mathbb{Q}\) is called modular if it is isomorphic to a quotient of the Jacobian \(J_1(N)_{\mathbb{Q}}\) of the modular curve \(X_1(N)_{\mathbb{Q}}\) for some integer \(N\). We further recall that all such are of \(\mathrm{GL}_2\)-type, which by definition means that \(\mathrm{End}_{\mathbb Q}(A) \otimes \mathbb Q\) contains a number field of absolute degree equal to the dimension of \(A\). The modularity of regular compatible systems implies the generalized Taniyama-Shimura-Weil conjecture: Any simple abelian variety over \(\mathbb Q\) which is of \(\mathrm{GL}_2\)-type is modular. The generalized Taniyama-Shimura-Weil conjecture extends the classical Taniyama-Shimura-Weil conjecture asserting the modularity of rational elliptic curves. For semistable rational elliptic curves, the latter statement had first been proved by A. Wiles [Ann. Math. (2) 141, No. 3, 443–551 (1995; Zbl 0823.11029)] and R. Taylor and A. Wiles [Ann. Math. (2) 141, No. 3, 553–572 (1995; Zbl 0823.11030)], thus giving a proof of Fermat’s Last Theorem. The general case of the classical Taniyama-Shimura-Weil conjecture was settled in [C. Breuil et al., J. Am. Math. Soc. 14, No. 4, 843–939 (2001; Zbl 0982.11033)].
Let \(\rho\colon \mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to \mathrm{GL}_2(\mathbb C)\) be an irreducible odd Galois representation. The modularity of irregular odd two-dimensional compatible systems of Galois representations over \(\mathbb Q\) implies that \(\rho\) arises from a newform of weight \(1\), which in turn proves Artin’s conjecture for this representation: The \(L\)-function attached to \(\rho\) admits an analytic continuation to the entire complex plane.
We next give an overview of the proof of Serre’s modularity conjecture, as presented in the article under review. The proof skillfully makes use of two main technical ingredients: a modularity lifting theorem requiring only weak local assumptions and a result embedding a representation of \(S\)-type into a compatible system of Galois representations upon which strong local conditions may be imposed. These two results are provided in the sequel to this article [loc. cit.] and are explained in some detail in the review of that article. With the further advancement of modularity lifting techniques, a more streamlined proof might be available in the future, like the one outlined in [C. Khare and J.-P. Wintenberger, Ann. Math. (2) 169, No. 1, 229–253 (2009; Zbl 1196.11076)].
In the article under review and its sequel the odd level case in odd characteristic and the weight two case in even characteristic of Serre’s modularity conjecture are proved. The cases left out are shown to be the consequence of a certain modularity lifting hypothesis, which is proved in [M. Kisin, loc. cit.].
To convey the main idea of the strategy of proof, we present a simplified version of it. The strategy consists of an induction having two main steps, namely ‘killing ramification in weight \(2\)’ and ‘reduction to weight \(2\)’. Start with a Galois representation \(\overline{\rho}\) of \(S\)-type. The aim is to prove that \(\overline{\rho}\) is modular. Suppose that the conductor \(N(\overline{\rho})\) is divisible by exactly \(r\) primes. By the step ‘reduction to weight \(2\)’ it suffices to prove the modularity of another representation \(\overline{\rho}_1\) of \(S\)-type such that \(N(\overline{\rho}_1)\) is still divisible by at most \(r\) primes but \(k(\overline{\rho}_1) = 2\). Next, the step ‘killing ramification in weight \(2\)’ reduces the modularity of \(\overline{\rho}_1\) to the modularity of yet another representation \(\overline{\rho}_2\) of \(S\)-type such that \(N(\overline{\rho}_2)\) is divisible by at most \(r-1\) primes. By proceeding with \(\overline{\rho}_2\) as with \(\overline{\rho}\) reduces the question inductively to the modularity of a level \(1\) representation, which is the first major result that Khare obtained on Serre’s modularity conjecture [C. Khare, Duke Math. J. 134, No. 3, 557–589 (2006; Zbl 1105.11013)].
The true strategy is complicated by the fact that both steps have some technical assumptions: oddness of the conductor, restriction to weight \(2\) in characteristic \(2\) and a locally good dihedral behaviour. An \(S\)-type representation \(\overline{\rho}\) is said to be locally good dihedral if it has a good dihedral prime \(q,\) which along with some congruence conditions on \(q\) means that \(\overline{\rho}\) restricted to a decomposition group at \(q\) is a dihedral group of order \(2 t\) with \(t\) a prime power satisfying certain extra conditions. The conditions ensure that the residual representations involved in the various steps of the proof have nonsolvable images. The nonsolvability is an important assumption for the modularity lifting theorem. The theorem embedding a residual representation into a compatible family is, in fact, devised such that a good dihedral prime can be introduced into the compatible family. This is referred to in the article as ‘raising the levels’, in analogy to the corresponding statement for modular representations. The notion of good dihedral prime turned out to be very useful also in different contexts, as for instance in the reviewer’s article [G. Wiese, in: Modular forms on Schiermonnikoog. Based on the conference on modular forms, Schiermonnikoog, Netherlands, October 2006. Edixhoven, Bas et al., Cambridge: Cambridge University Press, 343–350 (2008; Zbl 1217.12004)].
We now explain the basic strategy behind ‘killing ramification in weight \(2\)’. It uses the idea of ‘changing the prime’. Let \(\overline{\rho}\) be an \(S\)-type representation (assumed to be locally good dihedral) of weight \(2\) and residue characteristic \(p\) such that the level \(N(\overline{\rho})\) is divisible by at most \(r\) primes. It can be embedded into a strictly compatible system of Galois representations \((\rho_\iota)\) such that for each \(\iota\) the conductor \(N(\rho_\iota)\) is divisible only by primes that divide \(N(\overline{\rho})\). That \(p\) does not appear in the conductor is owed to the fact that \(\overline{\rho}\) is of weight \(2\). The conductor \(N(\overline{\rho}_s)\) of a residual representation at a prime \(s \mid N(\overline{\rho})\) is by definition away from \(s\) and only divisible by primes dividing \(N(\overline{\rho})\), thus by at most \(r-1\) primes. If \(s\) is not the good dihedral prime, then also \(\overline{\rho}_s\) is locally good dihedral. If \(\overline{\rho}_s\) is modular, then the modularity lifting theorem implies the modularity of the whole compatible system \((\rho_\iota)\), whence that of \(\overline{\rho}\).
For ‘reduction to weight \(2\)’ a similar strategy as in [C. Khare, op. cit.; Zbl 1105.11013] is applied, namely an induction on the residue characteristic. Let again \(\overline{\rho}\) be of \(S\)-type, \(N(\overline{\rho})\) divisible by at most \(r\) primes and of residue characteristic \(P\). The authors succeed to reduce the modularity of \(\overline{\rho}\) to the modularity of another Galois representation of residue characteristic \(p<P\) with at most \(r\) primes dividing the conductor, so that they can conclude inductively. In fact, the modularity in weight \(2\) is only needed for residue characteristics less than \(7\). Again, care is taken to ensure that all involved representations are locally good dihedral.