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

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 $\Pi$ be a cuspidal automorphic representation of $GL_{n}(\mathbb{A}_{L})$ which is conjugate self-dual. Assume that $\Pi_{\infty}$ is cohomological and not "slightly regular", as defined by Shin. In this case, Chenevier and Harris constructed an $l$-adic Galois representation $R_{l}(\Pi)$ 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 $R_{l}(\Pi)$ to the decomposition group at $v$ corresponds to the image of $\Pi_{v}$ via the local Langlands correspondence. We follow the strategy of Taylor-Yoshida, where it was assumed that $\Pi$ 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 $\Pi$ as above.Comment: 88 page

Multiplicative semigroups related to the 3x+1 problem

AbstractRecently Lagarias introduced the Wild semigroup, which is intimately connected to the 3x+1 conjecture. Applegate and Lagarias proved a weakened form of the 3x+1 conjecture while simultaneously characterizing the Wild semigroup through the Wild Number Theorem. In this paper, we consider a generalization of the Wild semigroup which leads to the statement of a Weak qx+1 Conjecture for q any prime. We prove our conjecture for q=5 together with a result analogous to the Wild Number Theorem. Next, we look at two other classes of variations of the Wild semigroup and prove a general statement of the same type as the Wild Number Theorem

Recent progress on Langlands reciprocity for GLn\mathrm{GL}_n: Shimura varieties and beyond

The goal of these lecture notes is to survey progress on the global Langlands reciprocity conjecture for $\mathrm{GL}_n$ over number fields from the last decade and a half. We highlight results and conjectures on Shimura varieties and more general locally symmetric spaces, with a view towards the Calegari-Geraghty method to prove modularity lifting theorems beyond the classical setting of Taylor-Wiles.Comment: 56 pages, to appear in the Proceedings of the 2022 IHES summer school on the Langlands progra

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 Mokrane's weight spectral sequence for log crystalline cohomology.Comment: 34 page

Patching and the p-adic local Langlands correspondence

We use the patching method of Taylor--Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL_n(F), F a finite extension of Q_p. We use our construction to prove many new cases of the Breuil--Schneider conjecture.Comment: Final version, to appear in Cambridge Journal of Mathematic

Components of moduli stacks of two-dimensional Galois representations

In a previous article we introduced various moduli stacks of two-dimensional tamely potentially Barsotti-Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil-Kisin modules with descent data. We study the irreducible components of these stacks, establishing in particular that the components of the former are naturally indexed by certain Serre weights.Comment: 71 pages. Comprises portions of the unpublished arXiv:1908.0701

The geometric Breuil-M\'ezard conjecture for two-dimensional potentially Barsotti-Tate Galois representations

We establish a geometrisation of the Breuil-M\'ezard conjecture for potentially Barsotti-Tate representations, as well as of the weight part of Serre's conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field.Comment: 24 pages. Comprises portions of the unpublished arXiv:1908.0701

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

We prove that the generic part of the mod l cohomology of Shimura varieties associated to quasi-split unitary groups of even dimension is concentrated above the middle degree, extending our previous work to a non-compact case. The result applies even to Eisenstein cohomology classes coming from the locally symmetric space of the general linear group, and has been used in joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Taylor and Thorne to get good control on these classes and deduce potential automorphy theorems without any self-duality hypothesis. Our main geometric result is a computation of the fibers of the Hodge-Tate period map on compactified Shimura varieties, in terms of similarly compactified Igusa varieties.Comment: 90 pages, accepted versio