Ana Caraiani

Local-global compatibility and the action of monodromy on nearby cycles

We strengthen the local-global compatibility of Langlands correspondences for GLn in the case when n is even and l 6D p. Let L be a CM field, and let ... be a cuspidal automorphic representation of GLn.AL/ which is conjugate self-dual. Assume that...1 is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an l-adic Galois representationRl..../ 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 Rl..../ to the decomposition group at v corresponds to the image of ...v via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that ... 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 strictly semistable schemes and we derive a generalization of the weight spectral sequence in this case. We also prove the Ramanujan–Petersson conjecture for ... as above.

Monodromy and local-global compatibility for l = p

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 \Pi\ 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 \Pi, 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 \Pi\ 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 Mokranes weight spectral sequence for log crystalline cohomology.

p-adic q-expansion principles on unitary Shimura varieties

We formulate and prove certain vanishing theorems for p-adic automorphic forms on unitary groups of arbitrary signature. The p-adic q-expansion principle for p-adic modular forms on the Igusa tower says that if the coefficients of (sufficiently many of) the q-expansions of a p-adic modular form f are zero, then f vanishes everywhere on the Igusa tower. There is no p-adic q-expansion principle for unitary groups of arbitrary signature in the literature. By replacing q-expansions with Serre–Tate expansions (expansions in terms of Serre–Tate deformation coordinates) and replacing modular forms with automorphic forms on unitary groups of arbitrary signature, we prove an analogue of the p-adic q-expansion principle. More precisely, we show that if the coefficients of (sufficiently many of) the Serre–Tate expansions of a p-adic automorphic form f on the Igusa tower (over a unitary Shimura variety) are zero, then f vanishes identically on the Igusa tower.This paper also contains a substantial expository component. In particular, the expository component serves as a complement to Hida’s extensive work on p-adic automorphic forms.

On the image of complex conjugation in certain Galois representations

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

On the generic part of the cohomology of compact unitary Shimura varieties

GL_n

Patching and the

over a totally real field

p

F

-adic local Langlands correspondence

.