Schedule for: 19w5210 - Modularity and Moduli Spaces

In my talk I survey some results obtained in recent years, notably by Kisin, Pappas and Zhu, on singularities arising in the reduction modulo $p$ of Shimura varieties with parahoric level structure. This relies on the theory of local models which are schemes defined in terms of linear algebra that model these singularities étale locally. Exemplary for some of the methods I explain a proof of their Cohen-Macaulayness which is joint work with Tom Haines

We will introduce monodromy local models and highlight some of their applications to Galois deformation theory and the Breuil-Mezard conjecture. We will focus on the role of a torus symmetry on these models. This is a report on work in progress with Le Hung, Levin, and Morra.

Let $G$ be a quasi-split reductive connected group over $\mathbb{Q}(t)$ which splits over $\mathbb{Q}(\zeta_e, t^{1/e})$, $e=2$ or $3$, whose derived group is absolutely simple simply connected and whose maximal torus corresponds to a sum of permutation modules of rank 1 or $e$. Fixing a maximal split torus $S$ of $G$ and a facet $\mathbf{f}$ of the apartment corresponding to $S$ in the building of $G$ over $\mathbb{Q}((t))$, we construct a smooth, affine and connected group scheme over $\mathbb{Z}[t]$, which should be regarded as a family of parahoric group schemes of type $\mathbf{f}$ in varying characteristics, generalising previous work of Pappas-Zhu over $\mathbb{Z}[1/e][t]$ and Tits over $\mathbb{Z}[t,t^{-1}]$. Since in the critical characteristic $e$ the group becomes generally pseudo-reductive, we briefly explain how Bruhat-Tits theory of reductive groups over local fields extends to the pseudo-reductive setting. Finally, we consider the local and global affine Grassmannians of the $\mathbb{Z}[t]$ group scheme and prove their representability by an ind-projective ind-scheme as well as normality of Schubert varieties. Time permitting, we discuss the resulting local models obtained in wildly ramified cases and their relation with the diamond local models of Scholze.

I will introduce the ordinary part (in the sense of Hida) of the higher coherent cohomology of Hilbert modular varieties and explain what we know about it: vanishing results and Hida style control theorems. This is joint work in progress with Vincent Pilloni.

Venkatesh and collaborators have introduced a variety of objects -- the local derived Hecke algebra, the global derived Hecke algebra, and the (global) derived Galois deformation ring -- in order to explain algebraic structures in the cohomology of locally symmetric spaces. I will review this story, and then introduce a new object that we call the "spectral Hecke algebra", which is a Hecke algebra that acts on the spectral side of Langlands, i.e. on moduli spaces of Galois representations. This is joint work with Akshay Venkatesh.

By work of Gee-Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D^*$ of $\mathrm{GL}(n)$ to a maximal compact subgroup of $\mathrm{GL}(n)$. Because of the congruence properties of the Jacquet-Langlands correspondence this transfer is compatible with the Breuil-Mézard formalism, which allows one to extend the Serre weight conjectures to $D^*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.

By construction, an eigenvariety comes with a map to the weight space. It is natural to ask what the ramification locus is. For Hilbert eigenvariety, we characterize the classical ramification points in terms of the associated Galois representation. This comes from the classicality theorem due to Tian-Xiao using cohomological methods, and a lower bound on the dimension of the tangent space of the weight fiber using Galois deformation. We would also mention how the ramification behaviors of the classical point and its companion points are related.

We will survey the known bounds on the eigenvalues of the $U_p$ Hecke operator appearing in various eigenvarieties. This will cover the speaker's work giving the polynomial growth rate of the associated Newton polygon for Chenevier's eigenvarieties for definite unitary groups of all ranks, as well as a theorem of Liu-Wan-Xiao and conjecture of Bergdall-Pollack giving more specific information for dimension-1 eigenvarieties.

We describe a Breuil--Mezard type conjecture that relates structure of mod $\ell$ fibers of deformation rings of the Galois group of a local field $K$ (which has residue characteristic $p$ different from $\ell$) to deformations of smooth mod $\ell$ representations of $\mathrm{GL}_n(K)$. These conjectures are centered around the compatibility between inertial local Langlands and its mod $\ell$ version. We give a proof of a weaker version of the conjecture, when $K $is a local function field, using the Taylor-Wiles-Kisin patching method. In the talk we will emphasize some feature and simplifications of the method in the function field setup. If time permits, we will report on some progress on variants for the conjecture for inner forms.

A key idea from Kisin's work on crystalline and semistable deformation rings involves constructing resolutions of these rings via moduli of Breuil-Kisin modules. For crystalline deformations with Hodge-Tate weights 0 or 1 the geometry of these resolutions closely models that of the deformation rings themselves, but for higher weights they are too large. I will explain a refinement of this approach which can be used to prove, for unramified extensions of $\mathbf{Q}_p$, potential diagonalisability of crystalline representations with weights $\le p$.

I will describe a method for determining the structure of a "patched module" arising from the Taylor-Wiles-Kisin method in the case when the local Galois deformation rings are not formally smooth. It was observed by Diamond that commutative algebra techniques (specifically the Auslander–Buchsbaum formula) can be used to show that a patched module is free, implying a mod $\ell$ multiplicity one statement, in the specific case when the relevant local deformation rings are all formally smooth. I will present a new method for determining the structure of certain patched modules which works when the local deformation rings are not formally smooth, provided these rings can still be computed explicitly, by exploiting the natural self-duality of many common patched modules. This method can be used to explicitly compute patched modules even in cases when they are not free. Using this (and Shotton's computations of local deformation rings in the $\ell \neq p$ case), I will compute the patched module arising from the cohomology of a Shimura curve, to prove a mod $\ell$ "multiplicity $2^k$" statement in the minimal level case, generalizing a result of Ribet. The precise computation of this patched module also yields additional information about the Hecke module structure of the cohomology of a Shimura curve, which among other things has applications to the study of congruence modules. Time permitting, I will also describe how this method can be extended to the "$l_0>0$" patching situation introduced by Calegari and Geraghty, and describe partial work towards generalizing it to the case of n dimensional representations associated to the cohomology of unitary Shimura varieties.

I will present new results about the representation theory of $p$-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at $p$. This simplifies earlier constructions of attaching Galois representations to automorphic representations for general linear groups (and general unitary groups). Moreover, our results apply to general reductive groups and have therefore the potential to become widely applicable beyond the case of the general linear group (and general unitary groups). This is joint work with Sug Woo Shin.