Even Galois representations and the Fontaine-Mazur conjecture
The author proves several theorems which would also be implied by the Fontaine-Mazur-conjecture (once justified). In particular, he proves the following: Let \(p>7\) be a prime number, \({\mathcal O}\) the ring of integers in some finite extension of \({\mathbb Q}_p\) and \(G\) the absolute Galois group of \(\mathbb Q\). Let \(\rho:G \to \text{GL}_2({\mathcal O})\) be a continuous irreducible representation which is unramified outside a finite set of primes. Assume furthermore that the restriction of \(\rho\) to a decomposition group of \(p\) is ordinary with distinct Hodge-Tate weights and that the image of the residual representation contains \(\text{SL}_2({\mathbb F}_p).\) Then for every complex conjugation \(c\) we find \(\det(\rho(c))=-1.\) In particular, there is no even Galois representation with different Hodge-Tate weights, and each representation as above is modular. The paper strongly relies on results of \textit{T. Barnet-Lamb, D. Geraghty, M. Harris} and \textit{R. Taylor} in [Publ. Res. Inst. Math. Sci. 47, No. 1, 29--98 (2011; Zbl 1264.11044)].
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