Euler系统的Eisenstein退化

希尔伯特-布卢门撒尔变量的爱森斯坦类

Les classes d'Eisenstein des varietes de Hilbert-Blumenthal

本文研究阿贝尔簇的Hilbert-Blumenthal族的Eisenstein类。我们首先给出了坐标表达式，这些一个在拓扑水平上，使用电流定义的莱文。然后，我们研究了这些Eisenstein类的退化在一个尖端的Baily-Borel紧化的Hilbert-Blumenthal品种。我们表明，使用显式描述的爱森斯坦类先前获得的，这些类退化的特殊值的$L$ -功能相关的基本完全实数域。然后，我们推导出几何证明克林根-西格尔定理和非零的结果，这些爱森斯坦类。

An Euler phi function for the Eisenstein integers and some applications

The Euler phi function on a given integer $n$ yields the number of positive integers less than $n$ that are relatively prime to $n$. Equivalently, it gives the order of the group of units in the quotient ring $\mathbb{Z}/(n)$. We generalize the Euler phi function to the Eisenstein integer ring $\mathbb{Z}[ρ]$ where $ρ$ is the primitive third root of unity $e^{2πi/3}$ by finding the order of the group of units in the ring $\mathbb{Z}[ρ]/(θ)$ for any given Eisenstein integer $θ$. As one application we investigate a sufficiency criterion for when certain unit groups $\left(\mathbb{Z}[ρ]/(γ^n)\rig…

Rankin--Eisenstein classes in Coleman families

We show that the Euler system associated to Rankin--Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in $p$-adic Coleman families. We prove an explicit reciprocity law for these families, and use this to prove cases of the Bloch--Kato conjecture for Rankin--Selberg convolutions.

Congruence Relations Connecting Tate-Shafarevich Groups with Bernoulli-Hurwitz Numbers by Elliptic Gauss Sums in Eisenstein Case

There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.

透过Eisenstein理想的视角

Through the lens of the Eisenstein ideal

本文通过Eisenstein理想建立模曲线几何与分圆域算术的联系，定义两个猜想互逆的映射，并给出其等变构造及Euler系统的应用。

反环分裂对角类与Beilinson--Flach元素

Anticyclotomic diagonal classes and Beilinson--Flach elements

本文比较了反环分裂对角循环Euler系统与环分裂Beilinson--Flach Euler系统，推广了Heegner点与Beilinson--Kato系统的联系，并推导出算术应用。

高斯整数与艾森斯坦整数的正则化乘积及素数

Regularized products of Gauss and Eisenstein integers and primes

受Muñoz García和Pérez-Marco工作的启发，采用欧拉式启发式计算，得到了高斯整数和艾森斯坦整数及其素数的正则化无穷乘积的显式表达式。

剩余可约 Hida 族中的 Iwasawa 不变量

Iwasawa invariants in residually reducible Hida families

研究在族与 Eisenstein 族相交情形下，尖点 Hida 族中模形式 $μ$-不变量的变化，特别关注穿过椭圆曲线 $X_0(11)$ 的 5-adic 族。