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Residually reducible representations and modular forms. (English) Zbl 1005.11030
We cite from the excellent introduction of this beautiful paper: In this paper we give criteria for the modularity of certain two-dimensional Galois representations. Originally conjectural criteria were formulated for compatible systems of $\lambda$-adic representations, but a more suitable formulation for our work was given by {\it J.-M. Fontaine} and {\it B. Mazur} [Elliptic curves, modular forms, and Fermat’s Last Theorem (Hong Kong, 1993), Internatational Press, Ser. Number Theory 1, 41-78 (1995; Zbl 0839.14011)]. Throughout this paper $p$ will denote an odd prime. Conjecture (Fontaine-Mazur): Suppose that $\rho: \text{Gal} (\overline{\bbfQ}/\bbfQ)\to \text{GL}_2(E)$ is a continuous representation, irreducible and unramified outside a finite set of primes, where $E$ is a finite extension of $\bbfQ_p$. Suppose also that (i) $\rho|_{I_p} \simeq \binom {*\ *}{0\ 1}$, where $I_p$ is an inertia group at $p$, (ii) $\det\rho= \psi\varepsilon^{k-1}$ for some $k\geq 2$ and is odd, where $\varepsilon$ is the cyclotomic character and $\psi$ is of finite order. Then $\rho$ comes from a modular form. To say that $\rho$ comes from a modular form is to mean that there exists a modular form $f$ with the property that $T(\ell)f= \operatorname {trace}\rho (\text{Frob}_\ell)f$ for all $\ell$ at which $\rho$ is unramified. Here $T(\ell)$ is the $\ell$th Hecke operator, and an arbitrary embedding of $E$ into $\bbfC$ is chosen so that $\operatorname {trace}\rho (\text{Frob}_\ell)$ can be viewed in $\bbfC$. Fontaine and Mazur actually state a more general conjecture where condition (i) is replaced by a more general, but more technical, hypothesis. The condition which we use, which we refer to as the condition that $\rho$ be ordinary, is essential to the methods of this paper. If we pick a stable lattice in $E^2$, and reduce $\rho$ modulo a uniformizer $\lambda$ of $O_E$, the ring of integers of $E$, we get a representation $\overline{\rho}$ of $\text{Gal} (\overline{\bbfQ}/\bbfQ)$ into $\text{GL}_2(O_E/\lambda)$. If $\overline{\rho}$ is irreducible, then it is uniquely determined by $\rho$. In general we write $\overline{\rho}^{ss}$ for the semisimplification of $\overline{\rho}$, and this is uniquely determined by $\rho$ in all cases. Previous work on this conjecture has mostly focused on the case where $\overline{\rho}$ is irreducible [cf. {\it A. Wiles}, Ann. Math. (2) 141, 443-551 (1995; Zbl 0823.11029); {\it F. Diamond}, Ann. Math. (2) 144, 137-166 (1996; Zbl 0867.11032)]. In that case the main theorems prove weakened versions of the conjecture under the important additional hypothesis that $\overline{\rho}$ has some lifting which is modular. This hypothesis, which is in fact a conjecture of Serre, is as yet unproved. In this paper we consider the case where $\overline{\rho}$ is reducible, and we prove the following theorem. Theorem. Suppose that $\rho: \text{Gal} (\overline{\bbfQ}/\bbfQ)\to \text{GL}_2(E)$ is a continuous representation, irreducible and unramified outside a finite set of primes, where $E$ is a finite extension of $\bbfQ_p$. Suppose also that $\overline{\rho}^{ss}\simeq 1\otimes \chi$ and that (i) $\chi|_{D_p}\ne 1$, where $D_p$ is a decomposition group at $p$, (ii) $\rho|_{I_p}\simeq \binom {*\ *}{0\ 1}$, (iii) $\det\rho= \psi\varepsilon^{k-1}$ for some $k\geq 2$ and is odd, where $\varepsilon$ is the cyclotomic character and $\psi$ is of finite order. Then $\rho$ comes from a modular form. We also prove similar but weaker statements when $\bbfQ$ is replaced by a general totally real number field. In the irreducible case the proof consists of identifying certain universal deformation rings associated to $\overline{\rho}$ with certain Hecke rings. However in the reducible case, even for a fixed $\overline{\rho}^{ss}\simeq 1\oplus\chi$, we have to consider all the deformation rings corresponding to the possible extensions of $\chi$ by 1. These deformation rings are not nearly as well-behaved as in the irreducible case. They are not in general equidimensional. Indeed there is a part corresponding to the reducible representations whose dimension grows with $\Sigma$, the finite set of primes at which we permit ramification in the deformation problem. Just as in the irreducible case, we do not know whether there is an irreducible lifting for each extension of $\chi$ by 1, but happily we do not need to assume this. In a previous paper [{\it C. Skinner} and {\it A. Wiles}, Proc. Natl. Acad. Sci. USA 94, 10520-10527 (1997; Zbl 0924.11044)] we examined some special cases where we could identify the deformation rings with Hecke rings. These cases roughly corresponded to the condition that there is a unique extension of 1 by $\chi$. In this paper we proceed quite differently. In particular we do not identify the deformation rings with Hecke rings. As we mentioned earlier, we consider the problem over a general totally real number field. This is not just to extend the theorem but is, in fact, an essential part of the proof. For it allows us by base change to restrict ourselves to situations where the part of the deformation ring corresponding to reducible representations has large codimension inside the full deformation ring. It should be noted that the base change we choose depends on $\Sigma$. We now give an outline of the paper. In § 2 we introduce and give a detailed analysis of certain deformation rings $R_{\cal D}$. These are associated to an extension $c$ of $\chi$ by 1. They are given as the universal deformation ring of the representation $$\rho_c: \text{Gal} (\bbfQ_\Sigma/\bbfQ)\to \binom {1\ *}{0\ \chi},$$ where the implied extension is given by $c$. Here $\bbfQ_\Sigma$ is the maximal extension of $\bbfQ$ unramified outside $\Sigma$ and $\infty$, although in the main body of the paper $\bbfQ$ is replaced by a totally real field $F$. In § 3 we give a corresponding detailed analysis of certain nearly ordinary Hecke rings introduced by Hida. We say that a prime of $R_{\cal D}$ is pro-modular if the trace of the corresponding representation occurs in a Hecke ring in a sense that is made precise in § 4. If all the primes on an irreducible component of $R_{\cal D}$ are pro-modular then we say that the component is pro-modular. If all the irreducible components of $R_{\cal D}$ are pro-modular then we say that $R_{\cal D}$ is pro-modular. The above theorem is deduced from our main result, which establishes the pro-modularity of $R_{\cal D}$ for suitable ${\cal D}$. There are three main steps in the proof of this latter result: (I) We show that if ${\germ p}$ is a “nice” prime of $R_{\cal D}$ then every component containing ${\germ p}$ is pro-modular. (II) We show that $R_{\cal D}$ has a nice prime ${\germ p}$. (III) We show that $R_{\cal D}$ is pro-modular. The proof of step (I) is modelled on that for the residually irreducible case and is given in §§ 5-8. The point is that the representation associated to $R_{\cal D}/{\germ p}$ is irreducible of dimension one and pro-modular. However the techniques of the irreducible case have to be modified, as this representation, which we now view as our residual representation, takes values in an infinite field of characteristic $p$. We should note also that the analog of the patching argument of {\it R. Taylor} and {\it A. Wiles} [Ann. Math. (2) 141, 553-572 (1995; Zbl 0823.11030)] is here performed on the deformation rings rather than on the Hecke rings. The proof of step (III) is given in Proposition 4.1. Steps (I) and (II) show that some irreducible component at the minimum level is modular. Then we use a connectivity result of M. Raynaud to show that there is a nice prime in every component at the minimum level. By step (I) again we deduce pro-modularity at the minimum level. A more straightforward argument then shows that there is a nice prime in every component of $R_{\cal D}$, so that we can again apply step (I) to deduce pro-modularity. For step (II) we proceed as follows. First we show, using the result of § 3.4 (which in turn uses techniques for proving the existence of congruences between cusp forms and Eisenstein series), that $R_{\cal D}$ has a nice prime for some extension $c_0$ of $\chi$ by 1. Using commutative algebra we show that there are primes in the subring of traces of $R_{\cal D}$ which correspond to representations with other reduction types, i.e. corresponding to a different extension $c$ (the pair 1, $\chi$ are fixed though). We make a construction to show that we can achieve all extensions in this way, and hence find nice primes for all extensions $c$. These primes are necessarily primes of the ring of traces which do not extend to $R_{\cal D}$ itself. The proof of step (II) is given in Proposition 4.2. At the start of the proof of this proposition is a more detailed outline of how we carry out step (II). We now briefly indicate the extra restriction in the case of a general totally real field $F$. We need to be able to make large solvable extensions of $F(\chi)$, the splitting field of $\chi$, with prescribed local behavior at a finite number of primes and such that the relative class number is controlled. When $F(\chi)$ is abelian over $\bbfQ$ we can do this using a theorem of Washington about the behavior of the $p$-part of the class number of $\bbfZ_\ell$-extensions. In the general case such a result is not known. Finally we note that the ordinary hypothesis, which is essential to our method, is frequently satisfied in applications. For example, suppose that $\rho$ (with $\overline{\rho}$ reducible) arises as the $\lambda$-adic representation associated to an abelian variety $A$ over $\bbfQ$ with a field of endomorphisms $K\hookrightarrow \text{End}_\bbfQ(A) \otimes \bbfQ$ such that $\dim A= [K: \bbfQ]$. Then the nearly ordinary hypothesis will hold provided $A$ is semistable at $p$, or even if $A$ acquires semistability over an extension of $\bbfQ_p$ with ramification degree $<p-1$. This can be verified by considering the Zariski closure of $\ker(\lambda)$ in the Neron model of $A$.

11F80Galois representations
11F11Holomorphic modular forms of integral weight
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