Number Theory Seminar

组织者 Organizer:
时间 Time: 每周一15:30-17:00,2019-9-9 ~ 12-30
地点 Venue:清华大学近春园西楼三层报告厅

讨论班简介 Description

报告摘要 Abstract


Title: Structure of p-adic period domains

Speaker: 陈苗芬 Miaofen Chen (East China Normal University)

Abstract: Rapoport and Zink introduce the p-adic period domain (also called the admissible locus) inside the rigid analytic p-adic flag varieties. Over the admissible locus, there exists a universal crystalline ℚp-local system which interpolates a family of crystalline representations. The weakly admissible locus is an approximation of the admissible locus in the sense that these two spaces have the same classical points. The Fargues-Rapoport conjecture for basic local Shimura datum gives a group theoretic characterization when the admissible locus and the weakly admissible locus coincide. In this talk, we will give a similar characterization for non-basic local Shimura datum. We will also discuss the question about where lives the weakly admissible points outside the admissible locus in general.


2019-12-30, 13:30-16:00, Conference room 1, Jin Chun Yuan West Bldg.

[1] 13:30-14:30

Title: Some old and new results about singular moduli

Speaker: Yingkun Li (Technische Universität Darmstadt)

Abstract: The values of the modular j-invariant at CM points are called singular moduli. They have been known to be algebraic integers since the time of Kronecker and Weber. Recently, Bilu-Habegger-Kühne showed that these are not units. In this talk, we will apply the results of Gross-Zagier, Gross-Kohnen-Zagier and their generalizations to give a short proof of this result.

[2] 15:00-16:00

Title: CM Value Formula for Orthogonal Shimura Varieties with Applications to Lambda Invariants

Speaker: Peng Yu 于鹏 (Morningside Center of Mathematics)

Abstract: In 1985, Gross and Zagier discovered a beautiful factorization formula for the norm of difference of singular moduli. This has inspired a lot of interesting work, one of which is the study of CM values of automorphic Green functions on orthogonal or unitary Shimura varieties. Now we generalize the definition of CM cycles beyond the ‘small’ and ‘big’ CM cycles and give a uniform formula in general using the idea of regularized theta lifts. Finally, as an application, we are able to give an explicit factorization formula for the norm of λ(½(d₁+d₁½)) - λ(½(d₂+d₂½)) with λ being the modular lambda invariant under the condition (d₁, d₂) = 1. The key observation is that λ(z₁) - λ(z₂) is a Borcherds product on X(2) × X(2).




Title: Stark-Heegner cycles for Bianchi modular forms

Speaker: Guhan Venkat (Morningside Center of Mathematics)

Abstract: In his seminal paper in 2001, Henri Darmon came up with a systematic construction of p-adic points, viz. Stark-Heegner points, on elliptic curves over the rationals. In this talk, I will report on the construction of local (p-adic) cohomology classes in the Harris-Soudry-Taylor representation associated to a Bianchi cusp form, building on the ideas of Henri Darmon and Rotger-Seveso. These local cohomology classes are conjecturally the restriction of global cohomology classes in an appropriate Bloch-Kato Selmer group and have consequences towards the Bloch-Kato-Beilinson conjecture as well as Gross-Zagier type results. This is based on a joint work with Chris Williams (University of Warwick).



Title: Some anabelian theorems for fields and curves

Speaker: Fucheng Tan 谭福成 (Research Institute for Mathematical Sciences, Kyoto University)

Abstract: This talk is an introduction to some well-known theorems in anabelian geometry. We will focus on Uchida’s theorem on function fields, and give a modern proof, i.e. an algorithm which reconstructs the function field group-theoretically via the Galois group of some solvably closed Galois extension. This is the so-called mono-anabelian approach. Time permitting, I will explain the absolute anabelian theorem for hyperbolic curves of strictly Belyi type.



Title: l-independence over Henselian valuation fields

Speaker: Weizhe Zheng (AMSS)

Abstract: A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne’s theorem to schemes of finite type over the ring of integers of a local field. This has applications to the l-independence of the l-adic cohomology of varieties over Henselian valuation fields, possibly of higher rank. This is joint work with Qing Lu.



Title: Towards the explicit description of local descent of symplectic supercuspidal representations of GL(2n)

Speaker: Jiajun Ma (Shanghai Jiaotong University)

Abstract: Descent is the "inverse" of Langlands functorial lift. The descent of symplectic supercuspidal representations of GL(2n) to SO(2n+1) over p-adic fields was established by Dihua Jiang, Chufeng Nien and Yujun Qin. In this talk, I will first give an explicit description of the local descent in the depth zero case. Then I will discuss a conjectural description in the general case. This is an ongoing joint work with Dongwen Liu, Chufeng Nien and Zhicheng Wang.



Title: The local Langlands correspondence for p-adic groups : construction of semi-simple Langlands parameters

Speaker: Laurent Fargues (Institut de mathématiques de Jussieu)

Abstract: I will present the general strategy to construct the local Langlands correspondence in the direction from representations to semi-simple Langlands parameters. This involves the stack of G-bundles on the curve. This is a joint work in progress with Peter Scholze.



Title: On the Yui-Zagier conjecture

Speaker: Tonghai Yang (University of Wisconsin, Madison)

Abstract: In the 1980s, Gross and Zagier discovered a beautiful factorization formula for norm of difference of singular moduli j(τ1)-j(τ2), where j is the famous j-invariants and τi are CM points of discriminants di<0. This was a test case for the well-known Gross-Zagier formula. They gave two proofs for the formula, algebraic one and analytic ones. Algebraic idea have been extended by Goren, Lauter, Viray, Howard and myself and others to the cases d1 and d2 not relatively prime and also to Hilbert modular surfaces. Analytic proof have been extended to Shimura varieties of orthogonal and unitary type using Borcherds’ regularized theta liftings, by Schofar, Bruinier, Kudla, myself, and others. In 1990s, Yui and Zagier made a similar but more subtle and surprising conjectural formula for norm of the difference of CM values of some Weber functions of level 48. In this talk, we will describe this conjectural formula and its proof using the so-called Big CM formula discovered by Bruinier, Kudla, and myself. This is joint work with Yingkun Li.



Title: Bessel functions and Beyond Endoscopy

Speaker: 齐治 Zhi Qi (Zhejiang University)

Abstract: In this talk, I will first introduce the thesis of Akshay Venkatesh on Beyond Endoscopy for Sym2 L-functions on GL2 over ℚ or a totally real field. The idea follows a suggestion of Peter Sarnak on using the Kuznetsov relative trace formula instead of the Arthur-Selberg trace formula for the Beyond Endoscopy problem. I will then discuss how to generalize Venkatesh’s work from totally real to arbitrary number fields. The main supplement is an integral formula for the Fourier transform of Bessel functions over ℂ.



Title: An automorphic descent construction for symplectic groups and applications

Speaker: 许宾 Bin Xu (Sichuan University)

Abstract: Automorphic descent, developed by Ginzburg-Rallis-Soudry, is a method which constructs concrete automorphic representations of classical groups, and has various applications in the study of automorphic representations. In this talk, we will introduce an automorphic descent construction for symplectic groups, and discuss its applications to global Gan-Gross-Prasad problem and quadratic twists of L-functions. This is a joint work with Baiying Liu.



Title: Regular supercuspidal representations and some applications

Speaker: Chong Zhang (Nanjing University)

Abstract: Regular supercuspidal representations are recently introduced by Kaletha, which are a subclass of tame supercuspidal representations. This new construction has many applications in the representation theory of reductive p-adic groups. In this talk, I will briefly review basic definition and properties of regular supercuspidal representations. I will also discuss the distinction problem for these representations, and also its relation with the local theta correspondence.



Title: Orientations of MW-Motives

Speaker: Nanjun Yang (YMSC, Tsinghua University)

Abstract: The category of (stable) MW-motives (defined by B. Calmès, F. Déglise and J. Fasel) is a refined version of Voevodsky's big motives, which provides a better approximation to the stable homotopy category of Morel and Voevodsky. A significant characteristic of this theory is that the projective bundle theorem doesn't hold.

In this talk, we introduce Milnor-Witt K-theory and Chow-Witt rings, which leads to the definition of (stable/effective) MW-motives over smooth bases. Then we discuss their quarternionic projective bundle theorem and Gysin triangles. As an application, we compute the Hom-groups between proper smooth schemes in the category of MW-motives.



Title: Acris-comparison of the Ainf-cohomology

Speaker: Zijian Yao 姚子建 (Harvard University)

Abstract: A major goal of p-adic Hodge theory is to relate arithmetic structures coming from various cohomologies of p-adic varieties. Such comparisons are usually achieved by constructing intermediate cohomology theories. A recent successful theory, namely the Ainf-cohomology, has been invented by Bhatt-Morrow-Scholze, originally via perfectoid spaces. In this talk, I will describe a simpler approach to prove the comparison between Ainf-cohomology and absolute crystalline cohomology, using the de Rham comparison and flat descent of cotangent complexes.



Title: Cycles on Shimura varieties via Geometric Satake

Speaker: Liang Xiao (Peking University)

Abstract: I will explain a joint work with Xinwen Zhu on constructing algebraic cycles on special fibers of Shimura varieties using geometric Satake theory. The talk will focus on explaining the key construction which upgrades the geometric Satake theory to a functor that relates the category of coherent sheaves on the stack [Gσ / G] to the category of sheaves on local Shtukas with cohomological correspondences as morphisms.




Title: Introduction to the GKZ-systems

Speaker: Jiangxue Fang (Capital Normal University)

Abstract: In this talk, I will review the theory of GKZ-systems discovered by Gelfand, Kapranov and Zelevinsky. In particular, I will study the composition series of GKZ-systems.



Title: Modularity and Cuspidality Criterions

Speaker: 王崧 Wang Song (中科院)

Abstract: We will survey cuspidality criterions for several cases of functoriality lifts for automorphic forms for $GL (N)$. Here is one important case we will sketch the proof: Let $\pi, \pi'$ are cuspidal automorphic representations for $GL (2), GL (3)$, and $\Pi = \pi \boxtimes \pi'$ the Kim-Shahidi lift from $GL (2) \times GL (3)$ to $GL (6)$. Then $\Pi$ is cuspidal unless two exceptional cases occur.  In particular, a modular form of Galois type which is associated to an odd icosahedral Galois representation must be cuspidal.



Title: The automorphic discrete spectrum of Mp(4)

Speaker: Atsushi Ichino (Kyoto University)

Abstract: In his 1973 paper, Shimura established a lifting from half-integral weight modular forms to integral weight modular forms. After that, Waldspurger studied this in the framework of automorphic representations and classified the automorphic discrete spectrum of the metaplectic group Mp(2), which is a nonlinear double cover of SL(2), in terms of that of PGL(2). We discuss a generalization of his classification to the metaplectic group Mp(4) of rank 2. This is joint work with Wee Teck Gan.



Title: Generalized zeta integrals on real prehomogeneous vector spaces

Speaker: 李文威Li Wenwei (北京大学)

Abstract: The Godement-Jacquet zeta integrals and Sato's prehomogeneous zeta integrals share a common feature: they both involve Schwartz functions and Fourier transforms on prehomogeneous vector spaces. In this talk I will sketch a common generalization in the local Archimedean case. Specifically, for a reductive prehomogeneous vector space which is also a spherical variety, I will define the zeta integrals of generalized matrix coefficients of admissible representations against Schwartz functions, prove their convergence and meromorphic continuation, and establish the local functional equation. Our arguments are based on various estimates on generalized matrix coefficients and Knop's work on invariant differential operators.




Title: Arithmetic of automorphic L-functions and cohomological test vectors

Speaker: 孙斌勇Sun Binyong (AMSS)

Abstract: It was known to Euler that $\zeta(2k)$ is a rational multiple of $\pi^{2k}$, where $\zeta$ is the  Euler-Riemann zeta function, and $k$ is a positive integer.  Deligne conjectured that similar results hold for motives over number fields, and automorphic analogue of Deligne's conjecture was also expected.  I will explain the automorphic conjecture, as well as some recent progresses on it. The Archimedean theory of cohomological representations and cohomological test vectors will also be explained, as they play a key role in the proof.



Title: Characteristic Cycles and Semi-canonical Basis

Speaker: 邓太旺Taiwang Deng (Max Planck institute for mathematics)

Abstract: Twenty years ago Lusztig introduced the semi-canonical basis for the enveloping algebra U(n), where n is a maximal unipotent sub-Lie algebra of some simple Lie algebra of type A, D, E. Later on B. Leclerc found a counter-example to some conjecture of Bernstein-Zelevinsky and related it to the difference between dual canonical basis and dual semi-canonical basis. He further introduced a condition (open orbit conjecture of Geiss-Leclerc-Schoer) under which dual canonical basis and dual semi-canonical basis coincide. In this talk we explain in detail the above relations and show a relation between the two bases above through micro-local analysis.



Title: Torsions in Cohomology of arithmetic groups and congruence of modular forms

Speaker: 邓太旺Taiwang Deng (Max Planck institute for mathematics)

Abstract: In this talk I will discuss the torsion classes in the cohomology of $SL_2(Z)$ as well as its variant with compact support. As a consequence, we show how to deduce congruences of cuspidal forms with Eisenstein classes modulo small primes. This generalizes the previous result on Ramanujan tau functions.




Title: Current methods versus expectations in the asymptotic of uniform boundedness

Speaker: Loïc Merel (Université de Paris)

Abstract: The torsion primes for elliptic curves over algebraic number fields of degree $d$ are bounded, according to the best current knowledge, exponentially in $d$. A disappointing result as polynomial bounds are expected. We will discuss what can be expected, and see how the use of the derived modular group can help clarify the limits of the current methods.



Title: Mathematical logic and its applications in number theory

Speaker: 任金波Jinbo Ren (University of Virginia)

Abstract: A large family of classical problems in number theory such as:

a) Finding rational solutions of the so-called trigonometric Diophantine equation $F(\cos 2\pi x_i, \sin 2\pi x_i)=0$, where $F$ is an irreducible multivariate polynomial with rational coefficients;

b) Determining all $\lambda \in \mathbb{C}$ such that $(2,\sqrt{2(2-\lambda)})$ and $(3, \sqrt{6(3-\lambda)})$ are both torsion points of the elliptic curve $y^2=x(x-1)(x-\lambda)$;

can be regarded as special cases of the Zilber-Pink conjecture in Diophantine geometry. In this talk, I will explain how we use tools from mathematical logic to attack this conjecture. In particular, I will present a series partial results toward the Zilber-Pink conjecture, including those proved by Christopher Daw and myself.

This talk is an expanded version of the one I gave during ICCM.



Title: Steenrod operations and the Artin-Tate Pairing

Speaker: Tony Feng (Stanford University)

Abstract: In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.



Title: Slopes of modular forms

Speaker: Bin Zhao (MCM, AMSS)

Abstract: In this talk, I will first explain the motivation to study the slopes of modular forms. It has an intimate relation with the geometry of eigencurves. I will mention two conjectures on the geometry of eigencurves: the halo conjecture concerning the boundary behavior, and the ghost conjecture concerning the central behavior. I will then explain some known results towards these conjectures. The former one is a joint work with Rufei Ren, which generalizes a previous work of Ruochuan Liu, Daqing Wan and Liang Xiao. The latter one is a joint work in progress with Ruochuan Liu, Nha Truong and Liang Xiao.



Title: Epsilon dichotomy for linear models

Speaker: Hang Xue (University of Arizona)

Abstract: I will explain what linear models are and their relation with automorphic forms. I will explain how to relate the existence of linear models to the local constants. This extends a classical result of Saito--Tunnell. I gave a talk last year here on the implication in one direction, I will explain my recent idea on the implication in the other direction.



Title: Quadratic twists of central L-values for automorphic representations of GL(3)

Speaker: Didier Lesesvre (Sun Yat-Sen University)

Abstract: A cuspidal automorphic representations of GL(3) over a number field, submitted to mild extra assumptions, is determined by the quadratic twists of its central L-values. Beyond the result itself, its proof is an archetypical argument in the world of multiple Dirichlet series, and therefore a perfect excuse to introduce these objects in this talk.



Title: Level-raising for automorphic forms on $GL_n$ over a CM field

Speaker: Aditya Karnataki (BICMR, Peking University)

Abstract: Let $E$ be a CM number field and $p$ be a prime unramified in $E$. In this talk, we explain a level-raising result at $p$ for regular algebraic conjugate self-dual cuspidal automorphic representations of $GL_n(\mathbf{A}_E)$. This generalizes previously known results of Jack Thorne.



Title: Curve counting and modular forms: elliptic curve case

Speaker: Jie Zhou (YMSC, Tsinghua University)

Abstract: In this talk, I will start by a gentle introduction of Gromov-Witten theory which roughly is a theory of the enumeration of holomorphic maps from complex curves to a fixed target space, focusing on the elliptic curve (as the target space) example. Then I will explain some ingredients from mirror symmetry, as well as a Hodge-theoretic description of quasi-modular and modular forms and their relations to periods of elliptic curves. After that I will show how to prove the enumeration of holomorphic maps are related to modular and quasi-modular forms, following the approach developed by Yefeng Shen and myself. Finally I will discuss the Taylor expansions near elliptic points of the resulting quasi-modular forms and their enumerative meanings. If time permits, I will also talk about some interesting works by Candelas-de la Ossa-Rodriguez-Villegas regarding the counting of points on and the counting of holomorphic maps to elliptic curves over finite fields.



Title: Integral period relations for base change

Speaker: Eric Urban (Columbia University)

Abstract: Under relatively mild  and natural conditions, we establish an integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the {\it congruence number} controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch-Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson-Flach element. This is a joint work with Jacques Tilouine.



Title: Geometry of Drinfeld modular varieties

Speaker: Chia-Fu Yu (Institute of mathematics, Academia Sinica)

Abstract: I will describe the current status on the geometry of Drinfeld moduli schemes we know. Main part of this talk will explain the construction of the arithmetic Satake compactification, and the geometry of compactified Drinfeld period domain over finite fields due to Pink and Schieder. We also plan to explain local and global properties of the strafification of reduction modulo v of a Drinfeld moduli scheme. This is joint work with Urs Hartl.