Numerical verification of Beilinson's conjecture for K2 of hyperelliptic curves

映射到阿贝尔簇的超椭圆曲线及其对零环的Beilinson猜想的应用

Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles

本文通过构造大量非对偶的超椭圆曲线双有理映射到阿贝尔曲面，并利用这些曲线在零环的Chow群中找到许多有理等价，从而推进了Beilinson猜想关于零环的证明。

Extensions of Motives and the Fundamental Group

In this paper we construct extensions of the Mixed Hodge structure on the fundamental group of a pointed algebraic curve. These extensions correspond to the regulator of certain explicit motivic cohomology cycles in the self product of the curve which were first constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. Our result is a generalization of a result of Colombo's where she constructs the extension corresponding to a motivic cycle class in the Jacobian of a hyperelliptic curve constructed by Collino. This is to appear in the Mathematical …

平面代数曲线上$\wp$-函数的计算

Computation of $\wp$-functions on plane algebraic curves

本文提出基于低亏格平面代数曲线黎曼面解析模型的$\wp$-函数数值计算方法，利用显式根式解直接计算Abel像，并用Jacobi反演验证，同时给出确定黎曼常数向量对应唯一特征的方法。

Isoharmonic deformations and constrained Schlesinger systems

We introduce and study the dynamics of Chebyshev polynomials on $d>2$ real intervals. We define isoharmonic deformations as a natural generalization of the Chebyshev dynamics. This dynamics is associated with a novel class of constrained isomonodromic deformations for which we derive the constrained Schlesinger equations. We provide explicit solutions to these equations in terms of differentials on an appropriate family of hyperelliptic curves of any genus $g=d-1\ge 2$. The verification of the obtained solutions relies on the combinatorial properties of the Bell polynomials and on the analysis…

二次多项式的动力学与亏格为$4$的曲线上的有理点

Dynamics of quadratic polynomials and rational points on a curve of genus $4$

本文在Flynn-Poonen-Schaefer猜想下，通过确定亏格4曲线上的有理点，证明二次多项式族中预周期点的有理参数集合大小的一致界，并给出Jacobian秩为1的解析与代数证明及强BSD猜想的数值验证。