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On a conjecture of Conrad, Diamond, and Taylor. (English) Zbl 1101.11017
The author proves a special case of a conjecture of C. Breuil and A. Mézard [Duke Math. J. 115, No. 2, 205–310 (2002; Zbl 1042.11030)], a part of which was already conjectured by B. Conrad, F. Diamond and R. Taylor [J. Am. Math. Soc. 12, No. 2, 521–567 (1999; Zbl 0923.11085)], hence the title.
Let $$p\neq2$$ be a prime number. Fix an algebraic closure $$\overline{\mathbb Q}_p$$ of $${\mathbb Q}_p$$ and denote by $$G_p$$ the absolute Galois group, by $$W_p$$ the Weil group, by $$I_p$$ the inertia subgroup and by $$\overline{\mathbb F}_p$$ the residue field. Let $$\overline{\mathbb Q}$$ be the algebraic closure $${\mathbb Q}$$ in $$\overline{\mathbb Q}_p$$.
Let $$E$$ be a finite extension of $${\mathbb Q}_p$$ in $$\overline{\mathbb Q}_p$$, with residue field $$F$$, and let $$\rho:G_p\rightarrow\text{GL}_2(F)$$ be a representation with “trivial endomorphisms”, i.e. the ring of $$F$$-linear $$G_p$$-equivariant endomorphisms of $$F^2$$ is reduced to $$F$$. Let $$\tau:I_p\rightarrow\text{GL}_2(E)$$ (the “type”) be a representation which extends to $$G_p$$ and has finite image. Finally, let $$k$$ (the “weight”) be an integer $$\geq2$$.
To the triple $$(k,\tau,\rho)$$, Breuil and Mézard attach two numbers, the “automorphic multiplicity” $$\alpha(k,\tau,\rho)$$ and the “Galois multiplicity” $$\gamma(k,\tau,\rho)$$.
$$\alpha(k,\tau,\rho)$$ : There is a unique (G. Henniart) $$\overline{\mathbb Q}_p$$-representation $$\sigma(\tau)$$ of $$\text{GL}_2({\mathbb Z}_p)$$ with finite image which occurs in the restriction to $$\text{GL}_2({\mathbb Z}_p)$$ of every ($$\infty$$-dimensional) irreducible admissible representation of $$\text{GL}_2({\mathbb Q}_p)$$ for which the restriction to $$I_p$$ of the associated representation of $$W_p$$ (R. P. Langlands’ correspondence) is isomorphic to $$\tau$$. Let $$a(m,n)$$ (with $$0\leq m\leq p-1$$ and $$0\leq n\leq p-2$$) be the multiplicity of the irreducible representation $$\text{ Sym}^m\overline{\mathbb F}_p^2\otimes_{\overline{\mathbb F}_p}\det^n$$ in the semisimplification of the reduction (to $$\overline{\mathbb F}_p$$) of $$\sigma(\tau)\otimes_{\overline{\mathbb Q}_p}\text{ Sym}{}^{k-2}\overline{\mathbb Q}_p^2$$. The number $$\alpha(k,\tau,\rho)$$ is essentially the sum, over those $$(m,n)$$ which occur (J.-P. Serre) in $$\rho|I_p$$, of the $$a(m,n)$$.
$$\gamma(k,\tau,\rho)$$ : Let $${\mathfrak o}$$ be the ring of integers of $$E$$. Consider deformations $$\tilde\rho:G_p\rightarrow\text{GL}_2(R)$$ of $$\rho$$ to complete local Noetherian $${\mathfrak o}$$-algebras $$R$$ with residue field $$F$$. Require that when $$R$$ is the ring of integers in a (totally ramified) finite extension $$L$$ of $$E$$, then $$\tilde\rho\otimes_RL$$ is potentially semistable of weights $$(0,k-1)$$, of determinant a fixed lift of $$\det(\rho)$$ which is the $$(k-1)$$-th power of the cyclotomic character times a character of finite order prime to $$p$$, and, finally, such that the restriction to $$I_p$$ of the representation of $$W_p$$ associated (J.-M. Fontaine’s theory) to $$\tilde\rho\otimes_RL$$ is equivalent to $$\tau$$. This deformation problem admits a versal solution $$R(k,\tau,\rho)$$; let $$\mathfrak M$$ be the maximal ideal of the local $$F$$-algebra $$R(k,\tau,\rho)\otimes_{\mathfrak o} F$$. Conjecturally, $$\dim_F{\mathfrak M}^n\!/\,{\mathfrak M}^{n+1}$$ stabilises for $$n\rightarrow+\infty$$. Admitting this, the number $$\gamma(k,\tau,\rho)$$ is this eventual dimension.
In their search for a “$$p$$-adic Langlands’ philosophy”, Breuil and Mézard made a deep conjecture of which the case “$$\det(\tau)$$ tame” says that $$\alpha(k,\tau,\rho)=\gamma(k,\tau,\rho)$$ [op. cit., Conjecture 2.3.1.1].
The present author proves the case “$$k=2$$ and $$\tau$$ tame” of their conjecture (Theorem 1.2). As a consequence, the versal deformation $${\mathfrak o}$$-algebra $$R(2,\tau,\rho)$$ (where $$\tau$$ is tame) turns out to be as predicted by Conrad, Diamond and Taylor [op. cit., Conjectures 1.2.2 and 1.2.3] (Theorems 6.22 and 6.23).
He follows the strategy of Breuil and Mézard in their proof of the case “$$k$$ even $$\leq p-1$$ and $$\tau$$ scalar” of their conjecture. To carry it out, he has to find “potential” versions, when $$k=2$$, of their machinery classifying lattices in semistable representations by means of “strongly divisible modules”.
Along the way, he classifies the possibilities for $$(\rho|_{I_p})\otimes_F\overline{\mathbb F}_p$$ when the representation $$\rho:G_p\rightarrow\text{GL}_2(F)$$ has trivial endomorphisms and is the reduction (to $$F$$) of a potentially crystalline representation $$G_p\rightarrow\text{GL}_2(E)$$ of weights $$(0,1)$$ (Corollary 6.15).
The main result has some striking applications to showing the modularity of odd representations $$\text{ Gal}(\overline{\mathbb Q}|{\mathbb Q}) \rightarrow\text{GL}_2(E)$$ unramified outside finitely many primes (cf. Theorem 1.6). Such theorems are going to be widely applicable, now that important new results about the modularity of odd representations $$\text{ Gal}(\overline{\mathbb Q}|{\mathbb Q}) \rightarrow \text{GL}_2(\overline{\mathbb F}_p)$$ (Serre’s conjecture) are becoming available (C. Khare, J.-P. Wintenberger). Indeed, results of this paper are used in their work.

##### MSC:
 11F80 Galois representations 14L15 Group schemes
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##### References:
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