Number Theory Group at Tsinghua University

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 thelocal 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 convergenceand 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.

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.

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 π, π′ be cuspidal automorphic representations for GL(2), GL(3), and Π = π ⊠ π′ the Kim-Shahidi lift from GL(2) × GL(3) to GL(6). Then Π 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.

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.

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.

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.

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.

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.

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.

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.

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 τ_i are CM points of discriminants d_i<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.

Let K be a finite extension of ℚ_p. A useful tool to study p-adic representations of G_K = Gal(K̅/K) is the theory of cyclotomic (φ,Γ)-modules of Fontaine, which relies on a characteristic 0 lift of the field of norms of the cyclotomic extension. In this talk, we will be interested in the following question: by what kind of Galois extensions K_∞/K can we replace the cyclotomic extension in order to build a (φ,Γ)-modules theory? We will show that under a certain additional assumption, such an extension is generated by the torsion points of a relative Lubin-Tate group and that the power series giving the action of the Galois group of K_∞/K are twists of semi-conjugates of endomorphisms of the same relative Lubin-Tate group.

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.

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.

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.

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.

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).

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 ℚ_p-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.

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.

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+d₁^½)) - λ(½(d₂+d₂^½)) 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).