Upcoming Seminar

Time：May 8, Wednesday, 16:00-17:00

Venue： Lecture Hall

Title: Drinfeld Lemma for F-isocrystals

Speaker： Daxin Xu 许大昕 (Chinese Academy of Sciences)

Abstract: Drinfeld's lemma for l-adic local systems is a fundamental result in arithmetic geometry. It plays an important role in the Langlands  correspondence for a reductive group over the function field of a curve over a finite field, pioneered by Drinfeld for GL_2 and subsequently extended by L. Lafforgue and then V. Lafforgue. In this talk, we will discuss Drinfeld's lemma for p-adic local systems: overconvergent/convergent F-isocrystals. This is based on a joint work with Kiran Kedlaya. 

Time：May 15, Wednesday, 16:00-17:00

Venue： Lecture Hall

Title: TBA

Speaker： Nadir Rafi Matringe (NYU Shanghai)

Abstract: TBA

Previous Seminars

Time：April 24, Wednesday, 16:00-17:00

Venue： Lecture Hall

Title: The global unramified geometric Langlands equivalence

Speaker： Lin Chen 陈麟 (Tsinghua University)

Abstract：Recently, Gaitsgory's school (to which I am honoured to belong) announced their proof of the global unramified geometric Langlands conjecture. I will explain the history, motivation and statement of this conjecture and the main ingredients used in this proof. If time permits, I will also introduce some questions in this field that remain open after this proof.

Time：April 10, Wednesday, 16:15-17:15

Venue： Lecture Hall

Title: Gelfand-Kirillov dimension in p-adic Langlands program

Speaker： Yongquan Hu 胡永泉 (Chinese Academy of Sciences)

Abstract：Gelfand-Kirillov dimension is an important concept in the study of admissible smooth representations of p-adic Lie groups. 

In this talk, I will explain how to control the Gelfand-Kirillov dimension for mod p representations coming from mod p cohomology in the case of GL_2. This is joint work with Breuil, Herzig, Morra, Schraen, and with Wang.

Time：April 10, Wednesday, 15:00-16:00

Venue： Lecture Hall

Title: How do generic properties spread?

Speaker： Yu Fu 付裕 (Caltech)

Abstract：Given a family of algebraic varieties, a natural question to ask is what type of properties of the generic fiber, and how those properties extend to other fibers. Let's explore this topic from an arithmetic point of view by looking at the scenario: Suppose we have a 1-dimensional family of pairs of elliptic curves over a number field $K$,  with the generic fiber of this family being a pair of non-isogenous elliptic curves. Furthermore, suppose the (projective) height of the parametrizer is less than or equal to $B$. One may ask how does the property of being isogenous extends to the special fibers. Can we give a quantitative estimation for the number of specializations of height at most $B$, such that the two elliptic curves at the specializations are isogenous? 

Time：March 6, Wednesday, 16:00-17:00

Venue： Lecture Hall

Title: The Brun--Titchmarsh Theorem

Speaker： Ping Xi 郗平(Xi’an Jiaotong University)

Abstract：It is fundamental to understand the distribution of primes in arithmetic progressions. With the aid of Brun’s sieve, Titchmarsh gave the first upper bound, which is of correct order of magnitude, on the number of such primes in an individual arithmetic progression. This gives the so-called Brun--Titchmarsh theorem. We will discuss our recent work on sharpening this theorem with better constants for general moduli and for special moduli, and the tools include Dirichlet polynomials, character/exponential sums, moments of L-functions, $\ell$-adic cohomology and spectral theory of automorphic forms. This is a joint work with Junren Zheng.

Time：January 3, Wednesday, 16:00-17:00

Venue： Lecture Hall

Title: Generating series of complex multiplication cycles

Speaker： Andreas Mihatsch (University of Bonn)

Abstract：Let $c_n$ be the number of isomorphism classes of pairs $(E, x)$  consisting of an elliptic curve $E$ over $\mathbb{C}$ and an  endomorphism $x$ that satisfies $x^2 = -n$. A classical theorem of  Zagier states that the series $\sum_{n = 1}^\infty c_n q^n$ is the  positive part of the $q$-expansion of a non-holomorphic modular form.  Its arithmetic version, due to Kudla--Rapoport--Yang, states that the  generating series of complex multiplication (CM) divisors on the  integral modular curve has a similar modularity property.  

In my talk, I will define CM cycle generating series for symplectic and  unitary Shimura varieties, and present first results on their  modularity. This adds a new facet to the Kudla program, which aims to  systematically relate special cycles on Shimura varieties with Fourier  expansions of automorphic forms. My talk is based on joint work with  Lucas Gerth, Siddarth Sankaran, and Tonghai Yang.