Potential automorphy and change of weight
The automorphy of Galois representations, as part of the Langlands program, is one of the most important problems in number theory. The present paper studies the potential automorphy of weakly compatible systems of \(l\)-adic representations of the absolute Galois group of a totally real number field. More precisely, let \(F\) be a number field, and \(G_F\) its absolute Galois group. A family of continuous semi-simple representations \[ r_\lambda : G_F\to\mathrm{GL}_n(\overline{M}_\lambda), \] indexed by finite places \(\lambda\) of a number field \(M\), is a weakly compatible system of \(n\)-dimensional \(l\)-adic Galois representations unramified outside a finite set \(S\) of places of \(F\) provided -- \(r_\lambda\) is unramified at all finite places \(v\not \in S\) for \(\lambda\) not dividing the residual characteristic of \(v\), and the characteristic polynomial of \(r_\lambda(\mathrm{Frob}_v)\) is a polynomial over \(M\) independent of \(\lambda\); -- \(r_\lambda\) is de Rham at all places \(v\) of \(F\) lying above the residual characteristic of \(\lambda\) and crystalline at all \(v\not\in S\) with that property; -- the Hodge-Tate numbers of \(r_\lambda\) are independent of \(\lambda\) for each embedding of \(F\) into the algebraic closure of \(M\). The main result of the paper is that a weakly compatible system, as defined above, is potentially automorphic if it is irreducible (all \(r_\lambda\) are irreducible), regular (every \(r_\lambda\) has \(n\) different Hodge-Tate numbers for any embedding of \(F\)), and odd essentially self-dual (\(F\) is totally real, \(r_\lambda\) all factor either through symplectic or orthogonal group with a parity condition on the multiplier character, and the multiplier characters form a weakly compatible system). For simplicity it is also assumed that the image of each embedding of \(F\) into the algebraic closure of \(M\) is actually contained in \(M\). Potential automorphy means that there is a finite Galois totally real extension \(F'\) of \(F\) such that all \(r_\lambda\) become automorphic over \(F'\). In particular, this means that the partial \(L\)-functions, given by the finite set \(S\), attached to complex Galois representations \(\iota(r_\lambda)\), where \(\iota\) is any embedding of \(M\) into \(\mathbb{C}\), all converge in some right half-plane and have the analytic continuation to a meromorphic function on the whole complex plane. This is a first such general potential automorphy theorem for dimension \(n>2\). Earlier results \textit{M. Harris} et al. [Ann. Math. (2) 171, No. 2, 779--813 (2010; Zbl 1263.11061)], \textit{T. Barnet-Lamb} et al. [Publ. Res. Inst. Math. Sci. 47, No. 1, 29--98 (2011; Zbl 1264.11044)], \textit{T. Barnet-Lamb} et al. [J. Am. Math. Soc. 24, No. 2, 411--469 (2011; Zbl 1269.11045)] are dealing with specific important examples. The proof of the main result relies on a higher-dimensional automorphy lifting theorem. The difficulties are in the fact that the automorphy lifting should work over a highly ramified base (since the motive used in the proof is usually over the extension \(F'/F\) which is highly ramified at \(l\)) and should incorporate a change of weight (since the \(l\)-adic cohomology of the motive used in the proof may be automophic of different weight than \(r_\lambda\)). The new automorphy lifting theorem proved in this paper overcomes these difficulties in some generality sufficient for the main result. The key new notion is potential diagonalizability of a potentially crystalline representation of the absolute Galois group of a local field. This notion allows to have the change of weight in the automorphy lifting theorem. Besides the main result, the automorphy lifting theorem has other applications. For example, if \(F\) is a CM field and \(\pi\) a regular, algebraic, essentially conjugate self-dual, cuspidal automorphic representation of \(\mathrm{GL}_n(\mathbb{A}_F)\), then, assuming \(\pi_\infty\) is of sufficiently regular weight, the \(n\)-dimensional \(l\)-adic representations associated to \(\pi\) are irreducible for \(l\) in a set of rational primes of Dirichlet density one. As a consequence of this paper and the work of Caraiani, the full local-global compatibility of the \(l\)-adic representations associated to such \(\pi\) over CM or totally real field can be obtained [\textit{T. Barnet-Lamb} et al., Ann. Fac. Sci. Toulouse, Math. (6) 21, No. 1, 57--92 (2012; Zbl 1259.11057)], [\textit{A. Caraiani}, Algebra Number Theory 8, No. 7, 1597--1646 (2014; Zbl 1310.11061)]. In a recent work [\textit{S. Patrikis} and \textit{R. Taylor}, ``Automorphy and irreducibility of some \(l\)-adic representations'', Compositio Math. (to appear)] proved a variant of the main result of this paper with the irreducibility assumption for \(r_\lambda\) replaced with purity. This may be more applicable in practice, because purity could be easier to check than irreducibility for compatible systems arising from geometry.
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