全局函数场的处处分歧塔

域扩张塔的不可分性指标

Indices of inseparability in towers of field extensions

设$K$是其剩余域具有特征$p$的局部域，$L/K$是$n=ap^ν$的有限可分全分歧扩张. $L/K$的不可分性指数$i_0,i_1,...,i_ν$由Fried在char $(K)=p$的情况下定义，由Heiermann在char $(K)=0$的情况下定义;它们给出了$L/K$的通常衍生数据的改进。不可分性指标可用于构造$0\le j\leν$的“广义Hasse-Herbrand函数”$ϕ_{L/K}^j$。本文给出了自然数$c$的值$ϕ_{L/K}^j(c)$的一个解释.我们使用这种解释来研究广义Hasse-Herbrand函数在域扩张塔中的行为。

类域塔的渐近增长模式

Asymptotic growth patterns for class field towers

本文研究 $\mathbb{Z}_p$-塔中数域上有限分歧伽罗瓦群的渐近增长模式，将其视为射线类群的非交换类似物，并在特定完全分裂条件下证明了精确的渐近下界，推广了 Iwasawa 关于类数增长的经典结果。

曲线 ${\mathbf Z}_p$-塔的斜率稳定性

Slope Stability for ${\mathbf Z}_p$-Towers of Curves

本文系统研究有限域上射影直线 $\mathbf Z_p$-塔的斜率稳定性，证明最小分歧断点假设下的部分斜率稳定性，并构造反例否定亏格稳定性蕴含斜率稳定性，给出不依赖常数域次数的改进界。

关于数域塔中类群的不变因子

On the invariant factors of class groups in towers of number fields

对于秩为d的有限阿贝尔p-群A，我们定义它的（对数）平均指数为它的基数的d次方根的以p为底的对数。研究了p-类群在数域塔中的平均指数的性质。通过将群论中的技巧与Brauer-Siegel定理的Tsfasman-Valdut推广相结合，我们构造了类群的平均指数保持有界的无穷多个tamely分歧塔.给出了几个p=2的显式例子。我们引入了一个附着在一个n阶生成的FAb pro-p群G上的不变量M（G），它度量了指数为n的子群的阿贝尔化的平均指数随着n趋于无穷大的渐近增长.当G=Gal（L/K）时，M（G）度量了类群在L/K中的平均指数的渐近性态.我们比较和对比分析与非解析群中的这种不变量的行为。我们利用这个不变量的相互作用的群论和数论的观点，并探讨了一些开放的问题，因此，这可能是独立的兴趣在群论。

Iwasawa Theory for $p$-torsion Class Group Schemes in Characteristic $p$

We investigate a novel geometric Iwasawa theory for $\mathbf{Z}_p$-extensions of function fields over a perfect field $k$ of characteristic $p>0$ by replacing the usual study of $p$-torsion in class groups with the study of $p$-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over $k$ associated to a $\mathbf{Z}_p$-extension of function fields totally ramified over a finite non-empty set of places, we investigate the growth of the $p$-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty$. By Dieudonné theory, this amounts to stud…

Iwasawa Dieudonné theory of function fields

Let $k$ be a perfect field of characteristic $p$ and $Γ$ an infinite, first countable pro-$p$ group. We study the behavior of the $p$-primary part of the "motivic class group", i.e. the full $p$-divisible group of the Jacobian, in any $Γ$-tower of function fields over $k$ that is unramified outside a finite (possibly empty) set of places $Σ$, and totally ramified at every place of $Σ$. When $Σ=\emptyset$ and $Γ$ is a torsion free $p$-adic Lie group, we obtain asymptotic formulae which show that the $p$-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $Σ\…

Lubin-Tate Deformation Spaces and Fields of Norms

We construct a tower of fields from the rings $R_n$ which parametrize pairs $(X,λ)$, where $X$ is a deformation of a fixed one-dimensional formal group $\mathbb{X}$ of finite height $h$, together with a Drinfeld level-$n$ structure $λ$. We choose principal prime ideals $\mathfrak{p}_n \mid (p)$ in each ring $R_n$ in a compatible way and consider the field $K'_n$ obtained by localizing $R_n$ at $\mathfrak{p}_n$, completing, and passing to the fraction field. By taking the compositum $K_n = K'_n K_0$ of each field with the completion $K_0$ of a certain unramified extension of $K'_0$, we obtain a…

Rankin-Selberg L-functions in cyclotomic towers, III

Let $π$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times \operatorname{GL}_2$ Rankin-Selberg $L$-functions associated to $π$ times representations induced from Hecke characters of $K$ which are ramified only at a given prime ideal $\mathfrak{p}$ of $F$. More specifically, we use spectral decompositions of shifted convolution sums and relations to Fourier-Whittaker coefficients of genuine and non-genuine metaplectic forms…

全局域的有限$p$-扩张上阿贝尔簇的$μ$-不变量变化

The $μ$-invariant change for abelian varieties over finite $p$-extensions of global fields

该研究拓展前人工作，分析全局域$\bZ_p^d$-扩张上常模阿贝尔簇的$μ$-不变量在有限Galois $p$-扩张下的变化，得到特征$p>0$时超奇异位处局部Galois上同调大小的显式界，给出所有特征下的渐近增长与不变量变化下界，并提供数值验证。

数域上的阿贝尔簇、驯服分歧与大伽罗瓦像

Abelian varieties over number fields, tame ramification and big Galois image

给定一个自然数n和数域K，我们证明了存在一个整数\ell_0，使得对任何素数\ell \geq \ell_0，存在一个有限扩张F/K，在\ell之上的所有地方都是非分歧的，以及F上一个n维的主极化阿贝尔簇A，使得所得到的\ell -挠表示ρ_{A，从G_F到GSp的\ell }（A[ \ell ]（\bar{F}））是满射的，处处光滑分歧.特别地，我们将GSp_{2n}（\mathbb{F}_\ ell）实现为数域F '/F的有限驯服扩张的伽罗瓦群，使得F在\ell上是非分歧的.

虚二次域的温和p类塔群

Mild p-Class Tower Groups of Imaginary Quadratic Fields

该研究针对虚二次域，通过定义范退化概型证明了864个p=3、203个p=5、全部3个p=7的相关数域的p类塔群为温和的，是首批被证明此类性质的数域。