贝尔数及相关序列的微分超越性：一种伽罗瓦理论方法

Finite index theorems for iterated Galois groups of cubic polynomials

Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational function of degree $d\geq 2$, and let $β\in\mathbb{P}^1(K)$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(β)=\text{Gal}(K(f^{-n}(β))/K)$ embed into $\text{Aut}(T_n)$, the automorphism group of the $d$-ary rooted tree of level $n$. A major problem in arithmetic dynamics is the arboreal finite index problem: determining when $[\text{Aut}(T_\infty):G_\infty]<\infty$. When $f$ is a cubic polynomial and $K$ is a function field of transcendence degree $1$ over an algebraic extension of $\mathbb{Q}$, we res…

Anabelian Intersection Theory I: The Conjecture of Bogomolov-Pop and Applications

We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let $F_{1}$ and $F_{2}$ be fields finitely-generated and of transcendence degree $\geq 2$ over $k_{1}$ and $k_{2}$, respectively, where $k_{1}$ is either $\bar{\mathbb{Q}}$ or $\bar{\mathbb{F}}_{p}$, and $k_{2}$ is algebraically closed. We denote by $G_{F_1}$ and $G_{F_2}$ their respective absolute Galois groups. Then the canonical map $φ_{F_{1}, F_{2}}: \Isom^i(F_1, F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1})$ from the isomorphisms, up to Frobenius twists, of the inseparable closures of $F_1$ and $F_2$ to continuous …

小抛物特征形式上Siegel亏格为2的模形式的辛Galois表示族

Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus $2$

本文构造了小抛物特征形式并研究其附带的辛Galois表示族，引入了带有G结构的(φ, Γ)-模和精炼的辛Galois表示族理论，并证明了在温和假设下的微分R=T定理，探讨了小抛物特征形式在Saito-Kurokawa提升与Bloch-Kato Selmer群的关系。

Hexponential 的几何与超越性

Geometry and Transcendence of the Hexponential

本文研究模群第二导出子群对应的六穿孔平面覆盖及其 hexponential 单值化映射，构造基于 Sturmian 序列的截面，证明其取值要么是 Markov 二次无理数要么是超越数，并给出 hexp 的连分数展开。

On deformation rings of residually reducible Galois representations and R=T theorems

We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation rho_0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents rho_1 and rho_2. Under some assumptions on Selmer groups associated with rho_1 and rho_2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaiche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Us…

关于与多项式递推相关的增长常数的超越性

On the transcendence of growth constants associated with polynomial recursions

本文去掉多项式首项系数相关有理数假设，研究多项式递推序列增长常数的超越性，证明其要么超越，要么其某幂次为Pisot数，还探讨了一类含数域元素的不等式解的问题。

限制在四分之一平面内的游走生成级数的长度导数

Length derivative of the generating series of walks confined in the quarter plane

本文利用差分伽罗瓦理论证明四分之一平面游走生成函数关于长度变量与关于空间变量满足代数微分关系的等价性，并据此刻画了79个经典游走模型的微分超越性。

非齐次一阶迭代函数方程及其在组合学中的应用

Inhomogeneous order 1 iterative functional equations with applications to combinatorics

本文研究一阶迭代函数方程的解，证明在特定条件下解要么是有理函数或一阶线性微分方程的解，要么是微分超越的，并将该结果应用于三类组合生成函数的微分超越性判定。

A sheaf-theoretic reformulation of the Tate conjecture

Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence. Our main result is that this conjecture implies other well-known conjectures in characteristic p, among which: - The (weak) Tate conjecture for smooth, projective varieties X over any finitely generated field of characteristic p: given a prime l di…

Annihilation of tor\_p(G\_S^ab) for real abelian extensions

Preprint of a paper to appear in "Communications in Advanced Mathematical Sciences". Let K be a real abelian extension of Q. Let p be a prime number, S the set of p-places of K and G\_K,S the Galois group of the maximal S-ramified pro-p-extension of K (i.e., unramified outside p and infinity). We revisit the problem of annihilation of the p-torsion group T\_K:=tor\_Z\_p(G\_K,S^ab) initiated by us and Oriat then systematized in our paper on the construction of p-adic L-functions in which we obtained a canonical ideal annihilator of T\_K in full generality (1978--1981). Afterwards (1992--2014) s…

数域上叶状结构的点计数

Point counting for foliations over number fields

本文证明数域上叶状结构叶与子簇交集的几何复杂性上界，并由此推导出关于同时挠点、特殊子簇及伽罗瓦轨道的多项式界，改进丢番图几何中的若干结果。