• Conference on Arithmetic Algebraic Geometry
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I will explain the definition and the basic comparison isomorphisms for prismatic cohomology, and use it to rederive some foundational results in perfectoid geometry. Joint work with Peter Scholze.

We will explain how the $A_{inf}$ cohomology of Bhatt, Morrow and Scholze can be used to compute the p-adic étale cohomology of Drinfeld's symmetric space in any dimension. This is joint work with Pierre Colmez and Wieslawa Niziol.

p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. This is joint work with Miaofen Chen and Xu Shen.

In joint work, we give a complete list of smooth and rationally smooth Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the quasi-minuscule Schubert variety in the quasi-split but non-split form of Spin8 (“ramified triality”) provides an input needed in the article by He-Pappas-Rapoport classifying Shimura varieties with good or semi-stable reduction. Our classification of smooth Schubert varieties verifies a conjectural classification of Rapoport.

After the work of Fargues-Fontaine,there is a good theory of Harder-Narasimhan filtrations for vector bundles on the Fargues-Fontaine curve.In this talk I'll discuss some existence results for "anti-Harder-Narasimhan" filtrations,where the slopes of the graded pieces go in the wrong direction. These results have applications to the geometry of the stack of vector bundles on the curve,which I'll also describe. This is joint work with Chris Birkbeck,Tony Feng, Serin Hong,Qirui Li, Anthony Wang, and Lynnelle Ye.

Let $F$ be a nonarchimedean local field and $\breve F$ be the completion of the maximal unramified extension of $F$. Let $G$ be a connected reductive group over $F$. The set $B(G)$ of the Frobenius-twisted conjugacy classes of $G(\breve F)$ is classified by Kottwitz. Motivated by Kottwitz's work, I introduced a decomposition of $G(F)$ into certain conjugate-invariant subsets, which are called the Newton strata. This decomposition has then found applications in the study of representations of $p$-adic groups, e.g. the Howe's conjecture and the trace Paley-Wiener theorem in the mod-$l$ setting. In this talk, I will explain the relation between the Newton maps over $F$ and over $\breve F$, and the relation between $B(G)$ and the Newton decomposition of $G(F)$. This is based on a recent joint work with S. Nie.

The Gross-Zagier formula expresses the Neron-Tate height of a Heegner point as the derivative of an L-function, while the Gross-Kohnen-Zagier formula expresses the Neron-Tate pairing of two different Heegner points as the product of the derivative of an L-function and a period integral. Yun and Zhang have proved a higher derivative version of the Gross-Zagier theorem, but with Heeger points on modular curves replaced by Heegner-Drinfeld cycles on moduli spaces of PGL(2)-Shtukas. I will describe a higher derivative version of Gross-Kohnen-Zagier for these Heegner-Drinfeld cycles. This is joint work with A. Shnidman.

The statements of the refined local and global Arthur-Langlands conjectures for reductive groups that are not quasi-split involve the cohomology of certain Galois gerbes. We shall review two constructions of such gerbes -- one due to Kottwitz and another due to us -- and discuss comparison results between the two in the local and global cases. We shall also discuss the implications of these comparison results to the local Langlands conjecture, the Kottwitz conjecture on the cohomology of Rapoport-Zink spaces, and the multiplicity formula for discrete automorphic representations. Some of this is joint work with J. Weinstein, D. Hansen, and O. Taibi.

Honda-Tate theory says that every abelian variety mod p is isogenous to the reduction of a CM abelian variety.We will discuss the analogous statement for isogeny classes on Shimura varieties, and explain what is conjectured and what is know.

In joint work with Jan Bruinier, Ben Howard, Michael Rapoport and Tonghai Yang, we proved that a certain generating series for the classes of arithmetic divisors on a regular integral model M of a Shimura variety for a unitary group of signature (n-1,1) for an imaginary quadratic field is a modular form of weight n valued in the first arithmetic Chow group of M. I will discuss how this is proved, highlighting the main steps. Key ingredients include information about the divisors of Borcherds forms on the integral model in a neighborhood of the boundary.

In this talk, I will explain why we have multiplicity one in the space of square integrable automorphic forms when the group is a unitary group and what is missing for the other classical groups to have a computation of such a multiplicity. This relies mainly on Arthur's theory and also on joint work with David Renard on the classification of the local components of the square integrable automorphic forms.

We determine the pro-étale cohomology of Drinfeld’s upper half space over a p-adic field K. The strategy is different from the one of Colmez, Dospinescu, Niziol, and is based on our approach describing global sections of equivariant vector bundles.

There is a well-established definition of a “canonical” integral model of a Shimura variety at unramified places of smooth reduction. This canonical model is characterized by a Neron type extension property. We will discuss the problem of extending a similar characterization to integral models of Shimura varieties at places where the level subgroup is parahoric, in which case their singularities are controlled by local models.

We will explain some results towards the construction of Hecke operators on the integral coherent cohomology of Shimura varieties and give arithmetic applications. This is work in progress with Najmuddin Fakhruddin.

We consider the Newton stratification on Iwahori double cosets in the loop group of a reductive group. In this context the Newton poset is the index set for non-empty Newton strata. We describe a group-theoretic condition on the double coset,called cordiality, under which the Newton poset is complete and Grothendieck’s conjecture on closures of Newton strata holds. We provide several large classes of examples for cordial double cosets.

The Chow ring of Shimura varieties is in general still very mysterious.But the tautological ring, defined as the subring generated by all Chern classes of all automorphic vector bundles, is a quotient of a purely combinatorial object and can there fore in principle be understood. On the other hand it contains many further interesting cycle classes. I will explain this and how to relate such classes for good reductions of Shimura varieties of Hodge type.

The Arithmetic Fundamental Lemma (AFL) conjecture arises from relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture. Recently some new conjectural AFLs have been discovered,including a linear AFL on Lubin-Tate spaces, and a unitary AFL on a unitary Rapoport-Zink space.For the linear AFL, Qirui Li has computed the intersection numbers.For the unitary AFL, I will explain its connection to a conjecture of Kudla-Rapoport on intersections of special divisors.

I’ll discuss the cohomology of moduli of Shtukas over elliptic Langlands parameters. In the case of GL(n), this recovers (and generalizes) L. Lafforgue’s result. In general, it resembles Arthur/Kottwitz' multiplicity formula. The proof does not make use of any trace formulas. Joint work with V. Lafforgue.