Robert Osburn
We use a result of Thaine to give an alternative proof of the fact that, for a prime p>3 congruent to 3 modulo 4, the component e_{(p+1)/2} of the p-Sylow subgroup of the ideal class group of \mathbb Q(ζ_{p}) is trivial.
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We define and apply a method to study the non-vanishing of $p$-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses a cyclotomic multiple harmonic sums as an infinite sum of products of $p$-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain $p$-adic cyclotomic multiple zeta values.
We prove that $μ= 0$ for the cyclotomic $\Z_p$-extensions of CM number fields.
For a newform for Gamma_0(N) of even weight k, we prove that its attached p-adic L-function is not identically zero on the group Z_p of the p-adic units. If p >3, we prove that the order of vanishing at any p-adic integer is finite.
Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups
本文证明分圆类群某些特征空间在相对密度为一的素数集合上消失或有界，支持 Vandiver 猜想与 Iwasawa 循环性猜想，并给出精确计算。
Odd vanishing cycles in cyclotomic fields
本文提出分圆域中Hecke群尖点的奇消没圈概念，刻画其轨道结构与对应单值群的性质，同时包含相关研究内容与综述内容。
For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.
Let $\K$ be a galois CM extension of $\Q$ and $\K_{\infty}$ its cyclotomic $\Z_p$-extension. Let $A_n$ be the $p$-parts of the class groups in the intermediate subfields $\K_n \subset \K_{\infty}$ and $\rg{A} = \varprojlim_n A_n$. We show that the $p$-rank of $\rg{A}$ is finite, which is equivalent to the vanishing of Iwasawa's constant $μ$ for $\rg{A}$. (Currently withdrawn)
Cyclotomic fields are generated by cyclotomic Hecke {\it L}-values of totally real fields, II
本文研究全实域上单个临界分圆 Hecke L 值能否生成相应分圆 Hecke 域，将 Jun-Lee-Sun 的平凡驯顺特征标结果推广到可解全实域上的非平凡 Hecke 特征标。
We improve an estimate of A.Granville (1987) on the number of vanishing Fermat quotients $q_p(\ell)$ modulo a prime $p$ when $\ell$ runs through primes $\ell \le N$. We use this bound to obtain an unconditional improvement of the conditional (under the Generalised Riemann Hypothesis) estimate of Y. Ihara (2006) on a certain sum, related to vanishing Fermat quotients. In turn this sum appears in the study of the index of certain subfields of of cyclotomic fields $\Q(\exp(2 πi/p^2))$.
Sharp Diameter Bounds for Nonnegative Cyclotomic Multiples
本文证明非负循环分圆倍式的支撑直径下界为(p-1)N/p，并刻画等式情形，证实Steinberger猜想，进而建立Coven-Meyerowitz直径界及其改进。
The local structure of refined class groups in cyclotomic Iwasawa theory
该研究给出分圆伊瓦萨理论中射线类精细结构的显式局部描述，证明了特定条件下 $\mathbb{Q}(\zeta_p)$ 的 $p$ 主射线类群分量非零的等价条件，导出伊瓦萨主猜想的射线类形式，恢复 Finat 公式并推导出精细 $\mu$ 消失结果。
The article is withdrawn. There is a gap in the proof of the vanishing of Iwasawa mu-invariant for the cyclotomic Z_p extension of a CM field, abelian over a totally real number field.
A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed prime, we prove the same proportion already among the primitive characters of any given order. The set of primitive characters modulo q of a given order can be described as an orbit under the action of the Galois group of the corresponding cyclotomic field. We also prove a positive proportion of nonvanishing within substantially shorter orbits generated by interm…
Bloch-regulator Principal Parts of Cyclotomic Iwasawa Pseudomeasures
本文构造分圆Iwasawa伪测度，研究其与布洛赫类、Coleman正则元的关系，推导权分量性质及纽结$5_2$的局部$K_3$-类准则。
Almost all primes are partially regular
该研究证明几乎所有素数在特定条件下满足部分规则性，通过Leopoldt反射得出部分Vandiver定理，并影响p-adic L-函数、模形式同余及K-群中的p- torsion。
Let $K$ be a number field. In the terminology of Nagell a unit $\varepsilon$ of $K$ is called {\it exceptional} if $1-\varepsilon$ is also a unit. The existence of such a unit is equivalent to the fact that the unit equation $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$ is solvable in units $\varepsilon_1,\varepsilon_2,\varepsilon_3$ of $K$. Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer $k$ with $k\geq 3$…
An important ingredient in the Ferrero--Washington proof of the vanishing of cyclotomic $μ$-invariant for Kubota--Leopoldt $p$-adic $L$-functions is an equidistribution result which they established using the Weyl criterion. The purpose of our manuscript is to provide an alternative proof by adopting a dynamical approach. A key ingredient to our methods is studying an ergodic skew-product map on $\mathbb{Z}_p \times [0,1]$, which is then suitably identified as a factor of the $2$-sided Bernoulli shift on the sample space $\{0,1,2,\cdots,p-1\}^{\mathbb{Z}}$.
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