北京雁栖湖应用数学研究院

Whittaker models have played a large role in the representation theory of reductive groups over finite, local and global fields. They are not available for all representations. There are the degenerate Whittaker models which are available more generally, usually studied when there are no Whittaker models, however, they have interest even when there is a Whittaker model. The lecture will be an exposition of some results, some due to me and others due to others.

It is a monumental achievement that Arthur proved the endoscopic classification theorem for automorphic representations of quasi-split symplectic and orthogonal groups. It was extended to quasi-split unitary groups by Mok. For non-quasi-split unitary groups, this is work in progress by Kaletha, Minguez, and myself (based on the earlier work together with White). All of these theorems are conditional on certain expected results, which are however highly nontrivial (at least to the speaker). I will discuss the current status. This talk is also based on joint work in progress with Hiraku Atobe, Wee Teck Gan, Atsushi Ichino, Tasho Kaletha, and Alberto Minguez.

I will first recall the general expectations of Shimura, Langlands, and Kottwtiz on the shape of the zeta function of a Shimura variety, or more generally its etale cohomology. I will then report on some recent progress which partially fulfills these expectations, for Shimura varieties of unitary groups and special orthogonal groups. Finally, I will give a preview of some foreseeable developments in the near future.

The $p$-adic Simpson correspondence aims to establish an equivalence between generalized representations and Higgs bundles over a $p$-adic variety. In this talk, we will explain how to upgrade such an equivalence to a twisted isomorphism of moduli stacks. This is based on a joint work in progress with Heuer.

Let $N$ and $p$ be prime numbers $\geq 5$ such that $p$ divides $N-1$. We prove a modulo $p$ version of the BSD conjecture at the $p$-Eisenstein ideal for even quadratic twists of $J_0(N)$. This is the analogue of a result of Mazur who considered odd quadratic twists. This is joint work with Jun Wang.

The integral solutions of Markoff surfaces has been studied by Markoff, Hurwitz, Mordell, ... etc, and more recently by Ghosh and Sarnak. In this talk, we will study integral points of Markoff surfaces from geometric point of view by using Brauer-Manin obstruction and prove that any Markoff surface does not satisfy strong approximation with Brauer-Manin obstruction. Part of this work has also been obtained by Loughran and Mitankin independently. This is a joint work with Colliot-Thelene and Dasheng Wei.

The idea of amplifications, dating back at least to van der Corput, Vinogradov, Kloosterman and Selberg, is quite elementary in the sense that one embeds the object of interests into a suitable family, and the individual information can be reflected on average. Such observations turns out to be very powerful in the developments of sieve methods, estimates for oscillatory sums and L-function theory. In this talk, we aim to give a short survey on the underlying ideas, and some recent applications to the above three typical objects will also be discussed.

As a generalization of Swan conductor T. Saito defined the characteristic cycle based on the construction of singular support by Beilinson. Its index formula was also shown by Beilinson and Saito. However, Grothendieck-Riemann-Roch type formula is left as a conjecture. In this talk, I wish to prove such a formula up to p-torsion.

This talk discusses the ongoing project to establish an equivalence between semi-stable local systems and log-prismatic F- crystals. A main new ingredient is a purity result, which allows us to reduce many problems in relative $p$-adic Hodge theory to the situation that the base ring is a complete discrete valuation ring (CDVR) with possibly non-perfect residue field. This is joint work with Heng Du, Yong Suk Moon, and Koji Shimizu.

I will discuss a logarithmic generalization of prismatic cohomology developed by Bhatt and Scholze. In particular, I will explain how it specializes and compares to other cohomology theories and the notion of Nygaard filtration. I will then discuss some ongoing projects on arithmetic applications of this theory. Part of this talk is based on my joint work with Teruhisa Koshikawa.

Let R be a field. Is the dimension of the restriction to the j-th congruence subgroup of a parahoric subgroup of an irreducible admissible R-representation of a $p$-adic reductive group G always given by the value at the j-th power of p of a polynomial with integer coefficients. I will explain why the answer is yes when p is odd and G=GL(2,Q_p), if the characteristic of R is p (Morra 2013), or when G is an inner form of GL(n,Q_p) or GL(n,F_p((t)), if the characteristic of R is not p (Henniart-Vigneras 2023).

I will discuss a conjectural categorical form of the local Langlands correspondence for $p$-adic groups and establish the unipotent part of such correspondence (for characteristic zero coefficient field). Joint work with Tamir Hemo.

I will explain how to understand geometrically Kaletha's gerb via the curve. Using this I define an extended Kottwitz set whose basic elements give all inner forms of a given $p$-adic group and that is moreover invariant under inner twisting. This allows us to state a geometrization conjecture for any reductive $p$-adic group."

Arthur established the local Langlands correspondence for $p$-adic classical groups, which parametrizes irreducible smooth representations by means of L-parameters. More recently, Fargues and Scholze attached an L-parameter to an irreducible smooth representation of an arbitrary $p$-adic reductive group. In this talk, I will compare these two L-parameters for some irreducible supercuspidal representations of Sp(6), including simple supercuspidal representations with trivial central character. This talk is based on a joint work with Masao Oi.

The Suppose that F is a totally real field in which p is inert. A (nice) 2-dimensional mod p representation of the absolute Galois group of F cuts out a mod p smooth representation pi of the $p$-adic group GL_2(F_p) in the cohomology of a Shimura curve. In previous work we constructed a functor D_A(-) from certain mod p smooth GL_2(F_p)-representations to multivariable (phi,Gamma)-modules, where Gamma = O_K^\times. We now show that D_A(pi) is isomorphic to D_A^\otimes(rho|_{G_{F_p}}), where D_A^\otimes(-) is a new functor from local Galois representations to multivariable (phi,Gamma)-modules, which we construct using perfectoid techniques. In particular, D_A(pi) depends only on the local Galois representation rho|_{G_{F_p}} at p. This is joint work with C. Breuil, Y. Hu, S. Morra, and B. Schraen.

The local Langlands correspondence predicts that irreducible smooth representations of $p$-adic groups can be naturally parametrized by Galois-theoretic data. There are now several approaches to constructing the local Langlands correspondence, and it is an important problem to reconcile them. I will explain what is currently known and highlight some interesting open questions.

On Shimura varieties, there are $p$-adic local systems which should be regarded as the $p$-adic etale realizations of universal G-motives. Lovering constructed crystalline realizations for integral models of Shimura varieties of abelian type at hyperspecial level. Recently Pappas-Rapoport constructed shtuka realizations. In this talk we discuss their prismatic refinements. This is a joint work with Hiroki Kato and Alex Youcis.

In 1981, Drinfeld enumerated the number of irreducible l-adic local systems of rank two on a projective smooth curve fixed by the Frobenius endomorphism. Interestingly, this number looks like the number of points on a variety over a finite field. Deligne proposed conjectures to extend and comprehend Drinfeld's result. In this talk, I will present Deligne's conjectures and discuss some mysterious phenomena that have emerged in various cases where this number is related to the number of stable Higgs bundles.

The congruence of modular forms is an important phenomenon in the arithmetic study of modular forms, or more generally, automorphic forms. For classical modular forms, many results have been obtained by Serre, Ribet, et al, for more than thirty years. In particular, Ribet used the arithmetic geometry of modular curves to find such congruence relation, also known as level raising. In an ongoing joint work with Yichao Tian and Liang Xiao, we generalize this phenomenon to higher-dimensional unitary Shimura varieties at inert places (which remains a conjecture in general), and its relation with a certain Ihara type lemma for such varieties. In the talk, I will explain cases for which we have confirmed such conjecture; and if time permits, we will mention its number-theoretical implications.