Monodromy and local-global compatibility forl=p

We strengthen the compatibility between local and global Langlands correspondences for GLₙ when n is even and l=p. Let L be a CM field and Π\ a cuspidal automorphic representation of GLₙ(AL) which is conjugate self-dual and regular algebraic. In this case, there is an l-adic Galois representation associated to Π, which is known to be compatible with local Langlands in almost all cases when l=p by recent work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was proved only up to semisimplification unless Π\ has Shin-regular weight. We extend the compatibility to Frobenius semisimplification in all cases by identifying the monodromy operator on the global side. To achieve this, we derive a generalization of Mokrane's weight spectral sequence for log crystalline cohomology.