完全剩余系统：一个引论及其应用

具有完美剩余域的混合特征完备离散赋值域的绝对伽罗瓦群之间同态的几何性质

Geometric properties of homomorphisms between the absolute Galois groups of mixed-characteristic complete discrete valuation fields with perfect residue fields

本文将 Mochizuki 与 Hoshi 关于 $p$-进局部域绝对伽罗瓦群几何同态的判别条件推广到具有完美剩余域的一般混合特征完备离散赋值域，并得到弱 Isom 远阿贝尔结果。

具有不完全剩余域的局部域的分枝

Ramification of local fields with imperfect residue fields

经典地，完备离散赋值域的伽罗瓦群的分歧滤子是在剩余域是完备域的情况下定义的。本文在不对剩余域作任何假设的情况下，定义了两个分歧滤子，并研究了它们的一些性质。

On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuat…

关于不完全剩余域情形下的p进Simpson与Riemann-Hilbert对应

On $p$-adic Simpson and Riemann-Hilbert correspondences in the imperfect residue field case

本文在剩余域具有有限p基的混合特征完全离散赋值域上，为Galois群的表示构造p进Simpson与Riemann-Hilbert对应，并由此证明Hodge-Tate与de Rham刚性定理，推广了Morita的结果。

数域中不完全分裂的最小素数

The smallest prime that does not split completely in a number field

研究数域中不完全分裂的最小素数的界问题。这是一个推广的经典问题的界定最小二次非剩余。在这里，我们提出了两种不同的方法来解决这个问题。第一个是通过研究1附近数域的戴德金ζ函数的行为，第二个是通过将问题与涉及乘性函数的问题联系起来。我们推导出最好的已知的界限，这个问题的所有数字领域的程度大于2。我们还得到了最有名的上界的残留的戴德金ζ函数的情况下，程度是小的判别式。

系数模p的乘积

Products of Factorials Modulo p

模p因子积与完全剩余的研究。

Multivariable $(φ,Γ)$-modules and Representations of Products of Galois Groups: The Case of Imperfect Residue Field

Let $K$ be a complete discretely valued field with mixed characteristic $(0, p)$ and imperfect residue field $k_K$. Let $Δ$ be a finite set. We construct an equivalence of categories between finite dimensional $\Bbb{F}_p$-representations of the product of $Δ$ copies of the absolute Galois group of $K$ and multivariable \' etale $(φ, Γ)$-modules over a multivariable Laurent series ring over $k_K$.

邀请到更高的局部域，第一部分，第13节：绝对非分歧完备离散赋值域的阿贝尔扩张

Invitation to higher local fields, Part I, section 13: Abelian extensions of absolutely unramified complete discrete valuation fields

本文介绍了作者关于具有特征p的任意剩余域的绝对非分歧完备离散赋值域K的循环p-扩张的理论.在这个理论中，从K的特征标群的p-部分到其剩余域上的Witt向量构造了一个同态.

带非完美剩余域的离散赋值域上$p$-代数的全分歧子域

Totally ramified subfields of $p$-algebras over discrete valued fields with imperfect residue

本研究针对特征$p$的完备离散赋值域上的$p$-代数，在剩余域$p$-秩满足一定条件下，证明了含全分歧纯不可分极大子域则必含全分歧循环极大子域的猜想，推进了相关代数结构研究。

具有不完备剩余域的$p$进局部域上阿贝尔簇的Faltings扩张与Hodge-Tate滤过

Faltings extension and Hodge-Tate filtration for abelian varieties over $p$-adic local fields with imperfect residue fields

针对剩余域不完备的特征0p进局部域，定义Z_p上O_K的Faltings扩张，推广Fontaine1981年的构造，为该类域上的阿贝尔簇构建Hodge-Tate滤过。

邀请到更高的局部域，第一部分，第15节：关于完备离散赋值域的Milnor K-群的结构

Invitation to higher local fields, Part I, section 15: On the structure of the Milnor K-groups of complete discrete valuation fields

这项工作包含了一个列表的所有已知结果的商过滤的Milnor K-群的一个完整的离散赋值领域的微分模的剩余域。利用指数映射和同伦复形研究了特征为0的弱分歧域与特征为p的剩余域的情形。

Partitioning the Primes and Efficient Non-trivial Factor Generation using the Least Odd Partition Identity in Conjunction with Dirichlet Linear Progressions and the Reduced Residue System Modulo 18

In this work we introduce the numerical constant, LOPI, N LOPI is congruent to LOPI mod 18, equal to the lowest odd partition identity in conjunction with the reduced residue system Modulo 18, a complete disjoint covering residue system when considered in its whole set of residues from 0 to 17. By convolution of specific LOPI Dirichlet linear progressions for each LOPI superset, the non-prime elements of the 6 reduced residue congruence classes mod 18 are generated sequentially and uniquely. Through set generation and complete composite number subset generation based on the pattern of sequenti…

Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible $\ell$-adic Galois Representation

Let $K$ be a complete, discretely valued field with finite residue field and $G_K$ its absolute Galois group. The subject of this note is the study of the set of positive integers $d$ for which there exists an absolutely irreducible $\ell$-adic representation of $G_K$ of dimension $d$ with rational traces on inertia. Our main result is that non-Sophie Germain primes are not in this set when the residue characteristic of $K$ is $> 3$. The result stated in the title is a special case.

Blocking Sets 与具有素因子个数上界的整数模的 Power Residue

Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors

本文建立了整数子集的 $q^{th}$ power residue 性质与有限几何中 $k$-blocking set 的等价关系，利用 Galois geometry 推导基数下界并刻画极小集合，证明该性质在 $\mathrm{PGL}(n,q)$ 作用下不变。

Cohen-Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields

Let $(R, \mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/\mathfrak{m} = \mathbb{F}_{q}$. Given a monic polynomial $P(t) \in R[t]$ whose reduction modulo $\mathfrak{m}$ gives an irreducible polynomial $\bar{P}(t) \in \mathbb{F}_{q}[t]$, we initiate the investigation of the distribution of $\mathrm{coker}(P(A))$, where $A \in \mathrm{Mat}_{n}(R)$ is randomly chosen with respect to the Haar probability measure on the additive group $\mathrm{Mat}_{n}(R)$ of $n \times n$ $R$-matrices. One of our main results generalizes two results of Friedman and Washington. O…

完备离散值域的Brauer p-维数

Brauer p-dimension of complete discretely valued fields

设K是特征为0的完备离散值域，其剩余域k的特征为p，n=[k：k^p]是k的p秩. Parimala和Suresh证明了K的Brauer p维数在n/2和2n之间。当n< 4时，我们将上界改进为n+1，并举例说明我们的上界是尖锐的。当n < 3时，我们还将下界改进为n。对于一般的$n$，我们构造了一个域族K_n，它具有p秩n的剩余域，使得K_n允许一个指数为p^{n+1}的中心单代数代数D_n.我们的尖锐下界为n<3和上界为n< 4结合这些例子的性质，促使我们推测，布劳尔p-维这样的领域总是在n和n+1之间。

完备超度量域的强u-不变量与周期-指标界

Strong u-invariant and Period-Index bound for complete ultrametric fields

本文研究完备超度量域的u-不变量与强u-不变量，通过剩余域描述之，并推广到函数域，同时给出Brauer-l-维数的界。

具有正特征的广义Hasse-Herbrand函数

Generalized Hasse-Herbrand functions in positive characteristic

设$L/K$是正特征的完备离散赋值域的扩张，并假定$K$的剩余域是完备的. $L$的剩余域不被假设为完美的。本文证明了广义Hasse-Herbrand函数$ψ_{L/K}^{\mathrm{ab}}$具有与经典Hasse-Herbrand函数相似的性质。特别地，我们证明了$ψ_{L/K}^{\mathrm{ab}}$是连续的，分段线性的，增加的，凸的，并满足一定的完整性性质。

根系zeta函数的通用极除子和旗标留数

Generic polar divisors and flag residues for root-system zeta functions

研究根系zeta函数，证明特定超平面是极除子，推导递归旗标留数等公式，恢复经典奇异数据及相关留数函数，确定$B_3$和$C_3$的一些性质，如双极点位置等。