Number Theory Group at Tsinghua University

In this talk, I will introduce a formula expressing the modular height of a quaternionic Shimura curve over a totally real field in terms of the logarithmic derivative at 2 of the Dedekind zeta function of the totally real field. The proof of the formula is inspired by the previous work of Yuan-Zhang-Zhang on the Gross-Zagier formula.

We first give an overview of recent results on Iwasawa theory and applications to BSD formulas for elliptic modular forms. Then we explain some ingredients used for the argument.

In his 1970 ICM report, Grothendieck asked the question to describe the p-adic analogues of Griffiths period domains. In this talk, we will review some constructions for these p-adic period domains, following recent developments in p-adic Hodge theory. We will then explain some ideas in a proof of the Fargues-Rapoport conjecture about the structure of certain p-adic period domains. This is joint work with Miaofen Chen and Laurent Fargues.

Let E be a CM elliptic curve defined over rational numbers with analytic rank one. The full BSD conjecture for E at a good supersingular prime is proved by Kobayashi, based on previous works by Gross-Zagier, Rubin, Kolyvagin, Perrin-Riou, et al. In the first talk I’ll briefly introduce these works and some key ingredients in them, and the comparison to my work at a potentially supersingular prime joint with Ye Tian.

Let E be a CM elliptic curve defined over rational numbers with analytic rank one. The full BSD conjecture for E at a good supersingular prime is proved by Kobayashi, based on previous works by Gross-Zagier, Rubin, Kolyvagin, Perrin-Riou, et al. In this second talk some more technical details will be concerned.

In 1987, Barry Mazur and John Tate formulated refined conjectures of the “Birch and Swinnerton-Dyer type”, and one of these conjectures was essentially proved in the prime conductor case by Ehud de Shalit in 1995. One of the main objects in de Shalit’s work is the so-called refined ℒ-invariant, which happens to be a Hecke operator. We apply some results of the theory of Mazur’s Eisenstein ideal to study in which power of the Eisenstein ideal ℒ belongs. One corollary of our study is the following elementary identity on supersingular j-invariants.

Let N be a prime number and p ≥ 5 be a prime dividing N − 1. For simplicity, assume N ≡ 1 (mod 12). Fix a surjective group homomorphism log: → ℤ/pℤ. Let S = {E₀, … , E_g} be the set of isomorphism classes of supersingular elliptic curves over . We denote by j(E_i) ∈ the j-invariant of E_i; it is well-known that j(E_i) ∈ . Let 𝒯(S) be the set of spanning trees of the complete graph with vertices in S. If T ∈ 𝒯(S), let E(T) be the set of edges of T. If 0 ≤ i ≠ j ≤ g, let [E_i, E_j] be the edge between E_i and E_j. We have:

In the first part of my talk, I shall review some definitions of Scholze’s pro-etale site and his proof of de Rham comparison theorem for rigid analytic spaces. Then, I shall concentrate in crystalline comparison theorems: I will introduce the so-called pro-etale crystalline topos, and sketch the proof of a version of crystalline comparison theorem, generalizing a previous result of Faltings. This part is based on a joint work with Yichao Tian.

Let V be a semi-stable non-crystalline p-adic Galois representation. By p-adic Hodge theory, one can associate to V the so-called Fontaine-Mazur L-invariants, which are invisible in the classical local Langlands correspondence. A task in p-adic Langlands program is to understand their counterpart on the automorphic side. In this talk, we will first review some of Breuil’s initial work on p-adic Langlands program and L-invariants in GL₂(ℚ_p)-case, and then report some recent progress (joint with Breuil) on higher L-invariants.

The Generalized Injectivity Conjecture of Casselman-Shahidi states that the unique irreducible generic subquotient of a (generic) standard module is necessarily a subrepresentation. It is related to L-functions, as studied by Shahidi, hence has some number-theoretical flavor, although our technics lie in the fields of representations of reductive groups over local fields. It was proven for classical groups (SO(2n+1), Sp(2n), SO(2n)) by M. Hanzer in 2010. In this talk, I will first explain our interest in this conjecture, and describe its main ingredients. I will further present our proof (under some restrictions) which uses techniques more amenable to prove this conjecture for all quasi-split groups.

R. Sharifi formulated remarkable conjectures which relate the arithmetic of cyclotomic fields to Eisenstein quotient of the homology groups of modular curves. In this talk, I will give a brief introduction to Sharifi’s conjectures and explain the statuses of these conjectures.