Behavior of the Torsion of the Differential Module of an Algebroid Curve under Quadratic Transformations

阿贝尔簇的挠率与扭形

Torsion and twists of abelian varieties

该研究探讨具好约化阿贝尔簇扭形的挠子群，证明相关二次扩张分歧素数定理，并指出满足Heegner假设的特定虚二次域上有理椭圆曲线无奇阶挠率增长。

Quadratic fields with a class group of large 3-rank

We prove that there are >>X^{1/30}/(log X) imaginary quadratic number fields with an ideal class group of 3-rank at least 5 and discriminant bounded in absolute value by X. This improves on an earlier result of Craig, who proved the infinitude of imaginary quadratic fields with an ideal class group of 3-rank at least 4. The proofs rely on constructions of Mestre for j-invariant 0 elliptic curves of large Mordell-Weil rank, and a method of the first author and Gillibert for constructing torsion in ideal class groups of number fields from rational torsion in Jacobians of curves. We also consider…

雅可比簇上的二次挠阶

Quadratic torsion orders on Jacobian varieties

研究有理数域上亏格\(g\geq2\)超椭圆曲线，通过引入特定多项式族方法，证明其雅可比簇存在特定阶有理挠点，确定了不同\(N\)值下超椭圆曲线与有理挠点的关系。

Cohen-Lenstra 启发式方法的一个几何途径

A geometric approach to the Cohen-Lenstra heuristics

本文提出一种几何方法，通过结式条件刻画二次域类群中元素的 $n$-挠性，并建立其与奇异亏格 $1$ 曲线 Selmer 群的联系，为 Cohen-Lenstra 启发式提供新视角。

分式理想的余长度与可约平面曲线的Tjurina数

Colengths of fractional ideals and Tjurina number of a reducible plane curve

本文针对复数域上可约代数胚平面曲线，通过建立分式理想余长度的易计算公式，精细化了Tjurina数公式，证明了Dimca猜想，并建立了Kähler微分模与Jacobian理想的联系。

具有有理j不变量的椭圆曲线在ℚ的极大初等阿贝尔2扩张上的挠率

Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$

本文针对定义在有理数域所有二次扩张合成域上、j不变量为非0或1728的有理数的椭圆曲线，完成了对其可能挠子群结构的分类。

A note on the ramification of torsion points lying on curves of genus at least two

Let $C$ be a curve of genus $g\geqslant 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\char(K)=0$ and that the characteristic of the residue field is not 2. Suppose that the Jacobian $\Jac(C)$ has semi-stable reduction over $R$. Embed $C$ in $\Jac(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique moderately ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\Jac(C)$.

Ranks of elliptic curves with prescribed torsion over number fields

We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is empty, or it contains curves of rank~0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group $T$ and a quartic field $K$ such that among the elliptic curves over $K$ with torsion subgroup $T$, there are curves of positive rank, but none of rank~0. We find examples of …

Fields of definition of elliptic curves with prescribed torsion

We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.

Surjectivity of the total Clifford invariant and Brauer dimension

Merkurjev's theorem--the statement that the 2-torsion of the Brauer group is represented by Clifford algebras of quadratic forms--is in general false when the base is no longer a field. The work of Parimala, Scharlau, and Sridharan proves the existence of smooth complete curves over local fields, over which Merkurjev's theorem is equivalent to the existence of a rational theta characteristic. Here, we prove that for smooth curves over a local field or surfaces over a finite field, replacing the Witt group by the total Witt group of all line bundle-valued quadratic forms recovers Merkurjev's th…