I will describe the main ideas that go into the proof of the (unramified, global) geometric Langlands conjecture. All of this work is joint with Gaitsgory and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, Lin, and Rozenblyum. I will also describe recent work on understanding the structure of Hecke eigensheaves (where the attributions are varied and too complicated for an abstract).
To any unital, associative ring R one may associate a family of invariants known as its algebraic K-groups. Although they are essentially constructed out of simple linear algebra data over the ring, they see an extraordinary range of information: depending on the ring, its K-groups can be related to zeta functions, corbordisms, algebraic cycles and the Hodge conjecture, elliptic operators, Grothendieck's theory of motives, and so on.
Our understanding of algebraic K-groups, at least as far as they appear in algebraic and arithmetic geometry, has rapidly improved in the past few years. This talk will present some of the fundamentals of the subject and explain why K-groups are related to the ongoing special year in p-adic arithmetic geometry. The intended audience is non-specialists.
p-adic heights have been a rich source of explicit functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical p-adic heights on abelian varieties from p-adic adelic metrics, using p-adic Arakelov theory developed by Besser. This construction closely mirrors Zhang's construction of canonical real valued heights from real-valued adelic metrics. We will use this new construction to give direct explanations (avoiding p-adic Hodge theory) of the key properties of p-adic height pairings needed for the quadratic Chabauty method for rational points.
In this talk I want to explain some surprising features of the pro-etale cohomology of rigid-analytic varieties, and how they can be explained by a six functor formalism with values in solid quasi-coherent sheaves on the Fargues-Fontaine curve.
Because of the existence of approximate p-power roots, a perfectoid algebra over ℚp admits no continuous derivations, and thus the natural Kahler tangent space of a perfectoid space over ℚp is identically zero. However, it turns out that many perfectoid spaces (or more general diamonds) arising from constructions involving rigid analytic varieties and their cohomology can be equipped with the extra structure of a Banach-Colmez Tangent Bundle, and using these Tangent Bundles natural period maps can be differentiated. As a first example, Tangent Bundles for infinite level basic EL local Shimura varieties have been computed by Ivanov and Weinstein using a heuristic construction due to Fargues and Scholze. However, in this talk, we are interested primarily in examples that go beyond the purview of the Fargues-Scholze heuristic. We will focus on the first non-trivial case of the infinite level modular curve, where we can describe the main points without assuming too much background with perfectoid rings or p-adic geometry. In this case, the existence of the Tangent Bundle is a consequence of an explicit quotient presentation of a closely related space derived from (a simple case of) the p-adic Simpson correspondence. Using this presentation, we will explain how to differentiate certain functions along Vector Fields in order to recover the annihilation property of Pan’s geometric Sen morphism, how to differentiate the Hodge-Tate period map, and how to relate this Tangent Bundle to a local construction via the Fargues-Scholze heuristic after restriction to good reduction residue disks. Time permitting, we will finish with a brief description of how the theory can be extended beyond infinite level modular curves.
The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group G. Most of the progress thus far has been concentrated on the case G = GL2, which has several special features. I will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). It is based on the intuition that the Breuil-Mezard conjecture is analogous to homological mirror symmetry.
Recent interactions between condensed mathematics and K-theory have led us to revisit the topic of (nonconnective) algebraic K-theory of topological algebras. In this talk, among recent developments, I will focus on the ring of continuous functions on a compact Hausdorff space valued in a local field (or a local division ring). This work resolves a previously unconfirmed claim about negative K-theory made by Rosenberg in 1990. The method employed is inspired by the resolution of Weibel's conjecture. The main result provides new counterexamples in K-theory by importing pathology from general topology.
I will explain what the question means and how to make it precise. Then I will give a conjectural answer. This is based on joint work with Peter Scholze.
With every bounded prism Bhatt and Scholze associated a cohomology theory of formal p-adic schemes. The prismatic cohomology comes equipped with the Nygaard filtration and the Frobenius endomorphism. The Bhatt-Scholze construction has been advanced further by Drinfeld and Bhatt-Lurie who constructed a cohomology theory with values in a stable ∞-category of prismatic F-gauges. The new cohomology theory is universal, meaning that, for every bounded prism, the associated prismatic cohomology theory factors through the category of prismatic F-gauges.
In this talk, I will explain how a full subcategory of the category of prismatic F-gauges formed by objects whose Hodge-Tate weights lie in the interval [0,p-2] is equivalent to the derived category of Fontaine-Laffaille modules with a similar weight constraint. In the geometric context, this means that the prismatic F-gauge associated with a formally smooth scheme over p-adic integers of dimension less than p-1 can be recovered from its Hodge filtered de Rham cohomology equipped with the Nygaard refined crystalline Frobenius endomorphism.
If time permits, I will explain a generalization of the above statement to the case of prismatic F-gauges over a smooth p-adic formal scheme.
Let K be a finite extension of ℚp. The Emerton-Gee stack for GL2 is a stack of etale (φ, Γ)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and can be viewed as a moduli stack of two-dimensional mod p representations of the absolute Galois group of K. By the work of Caraiani, Emerton, Gee and Savitt, it is known that in most cases, the locus of mod p representations admitting crystalline lifts with specified regular Hodge-Tate weights is an irreducible component of X. Their work relied on a detailed study of a closely related stack of etale phi-modules which admits a map from a stack of Breuil-Kisin modules with descent data. In our work, we assume K is unramfied and further study this map with a view to studying the loci of mod p representations admitting crystalline lifts with small, irregular Hodge-Tate weights. We identify these loci as images of certain irreducible components of the stack of Breuil-Kisin modules and obtain several inclusions of the non-regular loci into the irreducible components of X.
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