EVEN GALOIS REPRESENTATIONSAND THE FONTAINE-MAZUR CONJECTURE.II

摘要

be a continuous irreducible representation unramified away from finitely many primes. In [FM95], Fontaine and Mazur conjecture that if p is semi-stable at p, then either p is the Tate twist of an even representation with finite image or p is modular. In [Kis09], Kisin establishes this conjecture in almost all cases under the additional hypotheses that p Dp has distinct Hodge—Tate weights and p is odd (see also [Eme]). The oddness condition in Kisin's work is required in order to invoke the work of Khare and Wintenberger [KWO9a, KW09b] on Serre's conjecture. If p is even and p > 2, however, then P will never be modular. Indeed, when p is even and p Dp has distinct Hodge-Tate weights. the conjecture of Fontaine and Mazur predicts that p does not exist. In [Cal 11], some progress was made towards proving this claim under the additional assumption that p was ordinary at p. The main result of this paper is to remove this condition. Up to conjugation, the image of p lands in GL_2(O), where O is the ring of integers of some finite extension L/Qp (see Lemme 2.2.1.1 of [BM02]). Let F denote the residue field, and let p : GQ→ GL_2(F) denote the corresponding residual representation. We prove: