## Seminar Number Theory

## 摘要

## 报告人简介

**2019-11-15 Fri 13:00-14:30**

**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.

** **

**----------------------------History----------------------------**

**2019-11-11**

**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.

** **

**2019-11-4**

**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.

** **

**2019-11-1**

**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.

** **

**2019-10-21**

**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.

** **

**2019-10-15**

**Title:** A_{cris}-comparison
of the A_{inf}-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 A_{inf}-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 A_{inf}-cohomology
and absolute crystalline cohomology, using the de Rham comparison and flat
descent of cotangent complexes.

** **

**2019-10-14**

**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.

** **

** **

**2019-9-23**

**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.

** **

**2019-9-16**

**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.

** **

**2019-9-11**

**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.

** **

**2019-9-9**

**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.

** **

**-------------------------------History-------------------------------**

**2019-7-11**

**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.

** **

**2019-7-4**

**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.

** **

**2019-6-27**

**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.

** **

**2019-6-25**

**10:00-11:30**

**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.

** **

**13:30-15:00**

**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.

** **

**2019-6-24**

**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.

** **

**2019-6-6**

**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.

** **

**2019-5-30**

**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.

** **

**2019-5-23**

**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.

** **

**2019-5-16**

**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.

** **

**2019-4-4**

**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.

**2019-3-28**

**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.

**2019-3-21**

**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.