# Peking Online International Number Theory Seminar (POINTS) - 2022

## Introduction

This seminar series is sponsored by the Beijing International Center of Mathematical Research (BICMR) and the School of Mathematical Sciences of Peking University (SMS-PKU).

The seminars usually take place on Thursday mornings (UTC +8), presumably via Zoom. By default, the talks will be given in English. The official announcements of our talks will appear on the websites of BICMR and/or SMS-PKU. The talks will also be announced on researchseminars.org .

The videos, if available, can also be found on the channel of BICMR on Bilibili.com.

If you are willing to give a talk, please feel free to contact any one of the organizers below. Their E-mail addresses can be easily found on the websites of BICMR or SMS-PKU.

## Organizers

- Marc Besson
- Yiwen Ding
- Wen-Wei Li
- Ruochuan Liu
- Zhiyu Tian
- Liang Xiao
- Enlin Yang
- Xinyi Yuan

## Upcoming talks

To be scheduled.

## Past talks

### S=T for Shimura varieties

**Speaker:** Zhiyou Wu (BICMR)

**Time:** 13:30, September 21 (UTC+8)

**Abstract:** I will explain how the new \(p\)-adic geometry developed by Scholze can help prove the Eichler-Shimura relation for Shimura varieties of Hodge type, which has nothing to do with \(p\)-adic geometry a priori.

### A Fourier transform for Banach-Colmez spaces

**Speaker:** Arthur-César Le Bras (CNRS - IRMA Strasbourg)

**Time:** 15:00, September 14 (UTC+8)

**Abstract:** I will explain how to define an \(\ell\)-adic Fourier transform for Banach-Colmez spaces and discuss some examples. This is a joint work with Anschütz, which was motivated by the study of Fargues' geometrization conjecture for \(\mathrm{GL}_n\).

*Regular de Rham Galois representations in the completed cohomology of modular curves*

**Speaker:** Lue Pan (Princeton University)

**Time:** 09:30-11:00, August 30 (UTC+8)

**Abstract:** Let \(p\) be a prime. I want to explain how to use the geometry of modular curves at infinite level and Hodge-Tate period map to study \(p\)-adic regular de Rham Galois representations appearing in the \(p\)-adically completed cohomology of modular curves. We will show that these Galois representations up to twists come from modular forms and give a geometric description of the locally analytic representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\) associated to them. These results were previously known by totally different methods.

*Uniformly bounded multiplicities in the branching problem and D-modules*

**Speaker:** Masatoshi Kitagawa (Waseda University)

**Time:** 14:00-15:00, August 24 (UTC+8)

**Abstract:** In the representation theory of real reductive Lie groups, several finiteness results of lengths and multiplicities are known and fundamental. The Harish-Chandra admissibility theorem and the finiteness of the length of Verma modules and principal series representations are typical examples.

More precisely, such multiplicities and lengths are bounded on some parameter sets. T. Oshima and T. Kobayashi ('13 adv. math.) gave a criterion on which branching laws have (uniformly) bounded multiplicities.

In arXiv:2109.05556, I defined uniform boundedness of a family of \(\mathcal{D}\)-modules (and \(\mathfrak{g}\)-modules) to treat the boundedness properties uniformly. I will talk about its definition and applications. In particular, I will give a necessary and sufficient condition on uniform boundedness of multiplicities in the branching problem of real reductive Lie groups. Slides

*The local Gross-Prasad conjecture over archimedean local fields*

**Speaker:** Cheng Chen (University of Minnesota)

**Time:** 19:30-20:30, June 1 (UTC+8)

**Abstract:** The local Gross-Prasad conjecture is a refinement of the multiplicity one theorem for spherical pairs of Bessel type defined by a pair of special orthogonal groups. The conjecture shows that there is exactly one representation having multiplicity equal to one in each Vogan packet (with generic parameter) and it also depicts this unique representation with an epsilon character. I will introduce some recent progress for the conjecture over \(\mathbb{R}\) and \(\mathbb{C}\), part of the work was joint with Z. Luo. This local conjecture is a necessary ingredient for the global Gross-Prasad conjecture. Besides, the codimension-one case of the conjecture is closely related to the branching problem for special orthogonal groups. Slides Video

*Analytic continuation of branching laws for unitary representations*

**Speaker:** Jan Frahm (Aarhus Universitet)

**Time:** 15:00-16:30, May 18 (UTC+8)

**Abstract:** Branching problems ask for the behaviour of the restriction of an irreducible representation of a group \(G\) to a subgroup \(H\). In the context of smooth representations of real reductive groups, this typically leads to the study of multiplicities with which an irreducible representation of \(H\) occurs as a quotient of an irreducible representation of \(G\). Here, both quantitative results such as multiplicity-one theorems and qualitative results such as the Gan-Gross-Prasad conjectures are of interest.

In the context of unitary representations of real reductive groups, one can go a step further and explicitly decompose an irreducible representation of \(G\) into a direct integral of irreducible representations of \(H\). I will explain how branching laws for unitary representations are related to those in the smooth category, and how one can use an analytic continuation procedure along a principal series parameter to obtain explicit branching laws from certain Plancherel formulas for homogeneous spaces. Slides Video

*A Soergel bimodule approach to the character theory of real groups*

**Speaker:** Anna Romanov (University of New South Wales)

**Time:** 15:00, May 11 (UTC+8)

**Abstract:** Admissible representations of real reductive groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lustig—Vogan in the 80’s in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I’ll describe a categorification of a block of the LV module using Soergel bimodules.

*Number of irreducible representations in the cuspidal automorphic spectrum*

**Speaker:** Hongjie Yu (IST Austria)

**Time:** 15:30 April 27 (UTC+8)

**Abstract:** Let \(G\) be a reductive group defined and split over a global function field. We are interested in the sum of multiplicities of irreducible representations containing a regular depth zero representation of \(G(O)\), where \(O\) is the ring of integral adeles, in the automorphic cuspidal spectrum. The sum is expressed in terms of the number of \(\mathbb{F}_q\)-points of Hitchin moduli spaces of groups associated to \(G\). When \( G=GL(n) \), it implies some cases of Deligne's conjecture by Langlands correspondence.

*Prismatic approach to crystalline local systems*

**Speaker:** Haoyang Guo (Max-Planck-Institut für Mathematik)

**Time:** 15:30 April 20 (UTC+8)

**Abstract:** Let \(X\) be a smooth proper scheme over a \(p\)-adic field such that \(X\) has a good reduction. Inspired by the de Rham comparison theorem in complex geometry, Grothendieck asked if there is a "mysterious functor" relating étale cohomology of the generic fiber and crystalline cohomology of the special fiber. This question was answered by work of many people, including Fontaine and Faltings. In particular, this motivates the definition of a \(p\)-adic Galois representation being crystalline, generalizing the étale cohomology of \(X\) as above. In this talk, we will give an overview for the prismatic approach of Bhatt-Scholze on crystalline representations. Moreover, jointly with Emanuel Reinecke, we will consider the higher dimensional generalization of this approach on crystalline local systems.

*Pseudorandom Vectors Generation Using Elliptic Curves And Applications to Wiener Processes*

**Speaker:** Chung Pang Mok (Soochow University)

**Time:** 10:30-12:00 April 15 (UTC+8)

**Abstract:** Using the arithmetic of elliptic curves over finite fields, we present an algorithm for the efficient generation of sequence of uniform pseudorandom vectors in high dimension with long period, that simulates sample sequence of a sequence of independent identically distributed random variables, with values in the hypercube \( [0,1]^d \) with uniform distribution. As an application, we obtain, in the discrete time simulation, an efficient algorithm to simulate, uniformly distributed sample path sequence of a sequence of independent standard Wiener processes. Video

**arXiv:** 2201.00357

*Moduli spaces in p-adic non-abelian Hodge theory*

**Speaker:** Ben Heuer (Universität Bonn)

**Time:** 15:30 April 13 (UTC+8)

**Abstract:** In analogy to Simpson's non-abelian Hodge theory over the complex numbers, the p-adic Simpson correspondence over non-archimedean fields like \(C_p\) aims to relate p-adic representations of the étale fundamental group of a smooth proper rigid space \(X\) to Higgs bundles on \(X\). In this talk, I will introduce p-adic moduli spaces for either side of the correspondence, and explain how these can be compared by way of a non-abelian generalisation of the Hodge-Tate sequence. This allows one to construct new geometric incarnations of the p-adic Simpson correspondence, and to interpret the choices necessary for its formulation in a geometric fashion.

*On residues of certain intertwining operators*

**Speaker:** Sandeep Varma (Tata Institute of Fundamental Research)

**Time:** 10:30-11:30 April 6 (UTC+8)

**Abstract:** Let \(G\) be a connected reductive group over a finite extension \(F\) of \(\mathbb{Q}_p \). Let \(P = MN\) be a Levi decomposition of a maximal parabolic subgroup of \(G\), and \(\pi\) an irreducible unitary supercuspidal representation of \(M(F)\). One can then consider the representation \( \mathrm{Ind}_{P(F)}^{G(F)} \pi \) (normalized parabolic induction). Assume that \(P\) is conjugate to an opposite by an element \( w_0 \in G(F) \) that normalizes \(M\), and which fixes the isomorphism class of \(\pi \) (i.e., \( \pi \cong \,^{w_0}\pi \) ). Then, by the work of Harish-Chandra, \( \mathrm{Ind}_{P(F)}^{G(F)} \pi \) is irreducible if and only if a certain family \( A(s, \pi, w_0) \) of so called intertwining operators has a pole at \( s = 0 \). In this case, after making certain choices, the residue of \( A(s, \pi, w_0) \) at \( s = 0 \) can be captured by a scalar \( R(\tilde \pi) \in \mathbb{C} \), which has a conjectural expression in terms of some gamma factors related to Shahidi's local coefficients, as described by Arthur's local intertwining relation.

Following a program pioneered by Freydoon Shahidi, and furthered by him as well as David Goldberg, Steven Spallone, Wen-Wei Li, Li Cai, Bin Xu, Xiaoxiang Yu etc., one seeks to:

(a) get explicit expressions to describe \( R(\tilde \pi) \) ; and

(b) interpret the resulting expression for \( R(\tilde \pi) \) suitably, using the theory of endoscopy when applicable.

So far, these questions have been studied mostly for classical (including unitary) groups, or in some simple situations. We will discuss (a) above in a non-classical and slightly "less simple" situation, in the cases where \(G\) is an almost simple group whose absolute root system is of exceptional type or of type \(B_n\) with \(n \geq 3 \) or \(D_n \) with \( n \geq 4 \), and where \(P\) is a "Heisenberg parabolic subgroup". We will then comment on what we can say of (b) above in the \(G_2\), \(B_3\) and \(D_4\) cases. Though the reducibility results and the \( R(\tilde \pi) \) values are more or less already known in these cases by the Langlands-Shahidi method and related results (e.g., the work of Henniart and Lomeli and Caihua Luo in the case of \( D_4\) ), our investigations also suggest the existence of harmonic analytic expressions for certain gamma values, which in some cases just amount to the formal degree conjecture of Ichino, Ikeda and Hiraga, but in other cases seem slightly unwieldy and perhaps intriguing. Slides Video

*On arithmetic characterization of local systems of geometric origin*

**Speaker:** Alexander Petrov (Harvard University)

**Time:** 10:00 March 30 (UTC+8)

**Abstract:** I will talk about the problem of classifying local systems of geometric origin on algebraic varieties over complex numbers.

Conjecture: For a smooth algebraic variety \(S\) over a finitely generated field \(F\) , a semi-simple \(\mathbb{Q}_l\)-local system on \(S_{\bar{F}}\) is of geometric origin if and only if it extends to a local system on \(S_{F'} \) for a finite extension \(F' \supset F\) .

My main goal will be to provide motivation for this conjecture arising from the Fontaine-Mazur conjecture, and survey known results and related problems. Video

*On Deligne's conjecture for critical values of tensor product L-functions and symmetric power L-functions of modular forms*

**Speaker:** Shih-Yu Chen (Academia Sinica)

**Time:** 10:30-11:30 March 16 (UTC+8)

**Abstract:** In this talk, we introduce our result on the algebraicity of ratios of product of critical values of Rankin--Selberg \(L\)-functions and its applications. More precisely, let \(\mathit{\Sigma,\Sigma'}\) (resp. \(\mathit{\Pi,\Pi'}\) ) be cohomological tamely isobaric automorphic representations of \( \mathrm{GL}_n(\mathbb{A}) \) (resp. \(\mathrm{GL}_{n'}(\mathbb{A}) \) ) such that \( \mathit{\Sigma}_\infty = \mathit{\Sigma}_\infty' \) and \( \mathit{\Pi}_\infty = \mathit{\Pi}_\infty' \). It is a consequence of Deligne's conjecture on critical \(L\)-values that the ratio

\[

\frac{L(s, \mathit{\Sigma} \times \mathit{\Pi}) \cdot L(s,\mathit{\Sigma}' \times \mathit{\Pi}')}{L(s,\mathit{\Sigma} \times \mathit{\Pi}')\cdot L(s,\mathit{\Sigma}' \times \mathit{\Pi})}

\]

is algebraic and Galois-equivariant at critical points. We show that this assertion holds under certain parity and regularity conditions. As applications, we prove Deligne's conjecture for some tensor product \(L\)-functions and symmetric odd power \(L\)-functions for \( \mathrm{GL}_2 \). Video

*Algebraicity of critical values of automorphic L-functions: Examples and Conjectures*

**Speaker:** Shih-Yu Chen (Academia Sinica)

**Time:** 10:30-11:30 March 14 (UTC+8)

**Abstract:** In this talk, we introduce some algebraicity results on the critical values of automorphic \(L\)-functions. The techniques in these examples are integral representation of automorphic \(L\)-functions, constant terms of Eisenstein series, and their cohomological interpretations. These results are compatible with Clozel's conjecture on existence of motives associated to algebraic cuspidal automorphic representations of general linear groups and Deligne's conjecture on algebraicity of critical values of motivic \(L\)-functions. Video

*Mass formula on the basic loci of unitary Shimura varieties*

**Speaker:** Yasuhiro Terakado (NCTS)

**Time:** 10:30-12:00 March 3 (UTC+8)

**Abstract:** We study a mass of the group of self-quasi-isogenies of the abelian variety corresponding to a point on the basic locus in the reduction modulo p of a \(\mathrm{GU}(r,s)\) Shimura variety. We give explicit formulas for the number of irreducible components of the basic locus, and for the cardinality of the zero-dimensional Ekedahl-Oort stratum, in a Shimura variety associated with a unimodular Hermitian lattice. On the way, we also give a formula for the number of connected components of a Shimura variety. This is joint work with Chia-Fu Yu. Video

*The Jacquet-Zagier Trace Formula for GL(n)*

**Speaker:** Liyang Yang (Princeton University)

**Time:** 09:00-10:00 January 13, 2022 (UTC+8)

**Abstract:** The so-called Jacquet-Zagier trace formula was established by Jacquet and Zagier for GL(2) for two main reasons: deducing the holomorphy of adjoint L-functions and generalizing Selberg's trace formula in a different way from Arthur's truncation process. In this talk we will describe Jacquet-Zagier'strace formula in higher ranks. It plays a role in the study of holomorphic continuation of automorphic L-functions and certain Artin L-functions. Slides Video

*Endoscopic Relative Orbital Integrals on a Unitary Group*

**Speaker:** Chung-Ru Lee (Duke University)

**Time:** 10:00-11:00 January 6, 2022 (UTC+8)

**Abstract:** The characterization of distinguished representations is crucial for studying automorphic representations. The celebrated conjectures of Sakellaridis and Venkatesh provide such a characterization in many cases. In particular, they provide a conjectural description of the representations of a split reductive group that are distinguished by a split reductive spherical subgroup. However, there remain many mysteries when the generic stabilizer is disconnected.

The comparison of relative trace formulae, initially suggested by Jacquet, has been one of the most effective ways to study distinction problems in automorphic representation theory. Stabilization is a pivotal step for the comparison of relative trace formulae. To prepare for stabilization, one needs to investigate the endoscopic relative orbital integrals.

In this talk, we study the endoscopy theory for unitary groups in a relative setting where the generic stabilizer is disconnected and finite over a p-adic field. This talk aims to compute an explicit formula for endoscopic relative orbital integrals. Slides Video