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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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