arXiv subjects
Publications and source records attributed to Ana Caraiani.
We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $GL_n$ over a totally real field $F$.
We strengthen the compatibility between local and global Langlands correspondences for GL_{n} when n is even and l=p. Let L be a CM field and Π a cuspidal automorphic representation of GL_{n}(\mathbb{A}_{L}) 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.
We strengthen the local-global compatibility of Langlands correspondences for $GL_{n}$ in the case when $n$ is even and $l\not=p$. Let $L$ be a CM field and $Π$ be a cuspidal automorphic representation of $GL_{n}(\mathbb{A}_{L})$ which is conjugate self-dual. Assume that $Π_{\infty}$ is cohomological and not "slightly regular", as defined by Shin. In this case, Chenevier and Harris constructed an $l$-adic Galois representation $R_{l}(Π)$ and proved the local-global compatibility up to semisimplification at primes $v$ not dividing $l$. We extend this compatibility by showing that the Frobenius semisimplification of the restriction of $R_{l}(Π)$ to the decomposition group at $v$ corresponds to the image of $Π_{v}$ via the local Langlands correspondence. We follow the strategy of Taylor-Yoshida, where it was assumed that $Π$ is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator $N$ on the complex of nearby cycles on a scheme which is locally etale over a product of semistable schemes and derive a generalization of the weight-spectral sequence in this case. We also prove the Ramanujan-Petersson conjecture for $Π$ as above.