On a conjecture of Conrad, Diamond, and Taylor
The author proves a special case of a conjecture of \textit{C.~Breuil} and \textit{A.~Mézard} [Duke Math. J. 115, No. 2, 205--310 (2002; Zbl 1042.11030)], a part of which was already conjectured by \textit{B. Conrad, F. Diamond} and \textit{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 \(\alpha(k,\tau,\rho)=\gamma(k,\tau,\rho)\) [\textit{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 [\textit{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.
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