## Seminar Number Theory

## 报告人简介

**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.