Number Theory Group at Tsinghua University

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.

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 congruence number controlling the congruence 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.

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.

Let E be an elliptic curve over the rationals with conductor N. A conjecture due to Perrin-Riou describes a rank one aspect of the associated p-adic Beilinson-Kato elements for a prime p. In the case (p,2N)=1, we plan to describe setup and strategy in our result towards the conjecture (joint with Christopher Skinner and Ye Tian).

In this talk, we will explain Andreatta-Iovita’s work on the p-adic iteration of Gauss-Manin connections in the space of nearly overconvergent elliptic modular forms (of unbounded order) and its application to the construction of triple product p-adic L-functions. If time permits, we will explain how to generalize these results to rational Shimura curves.

As an overview of Arakelov geometry, we first introduce several Riemann-Roch problems for arithmetic K₀-groups. They deduce various interesting structural and quantitative statements which relate arithmetical invariants of algebraic varieties to special values of logarithmic derivatives of L-functions. Then we turn to higher arithmetic K-theory, we establish its covariant functoriality by using analytic torsion for cubes of hermitian vector bundles.

Let 𝔽_q be a finite field with q elements and odd characteristic. A pair (G, G′) of mutually centralized reductive subgroups of Sp_2n(𝔽_q) is called a reductive dual pair. By means of the Weil representation of Sp_2n(𝔽_q), Roger Howe introduced a correspondence Θ: ℛ(G) → ℛ(G′) between the category of complex representations of these subgroups. Here we discuss how this correspondence relates the Harish-Chandra series of G to those of G′. If time allows, we will discuss how this correspondance can be expressed as a correspondance between unipotent Harish-Chandra series.

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(𝔸_E). This generalizes previously known results of Jack Thorne.

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.

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.

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.

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.

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.

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

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

b) Determining all λ∈ℂ such that (2, (2(2-λ))^1/2) and (3, (6(3-λ))^1/2) are both torsion points of the elliptic curve y²=x(x-1)(x-λ);

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.

In this talk I will discuss the torsion classes in the cohomology of SL₂(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.

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.

It was known to Euler that ζ(2k) is a rational multiple of π^2k, where ζ 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.