革兰氏定律在一定时间内失效

A positive proportion of Thue equations fail the integral Hasse principle

For any nonzero $h\in\mathbb{Z}$, we prove that a positive proportion of integral binary cubic forms $F$ do locally everywhere represent $h$ but do not globally represent $h$; that is, a positive proportion of cubic Thue equations $F(x,y)=h$ fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms $F$ by their absolute discriminants. We prove the same result for Thue equations $G(x,y)=h$ of any fixed degree $n \geq 3$, provided that these integral binary $n$-ic forms $G$ are ordered by the maximum of the absolute values of their coefficients.

A Positive Proportion of Hasse Principle Failures in a Family of Châtelet Surfaces

We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bretèche and Browning.

正比例的平面三次函数不符合哈塞原理

A positive proportion of plane cubics fail the Hasse principle

当$\mathbb Z$上的所有三元三次型都按其系数的高度排序时，我们证明了它们中的正比例不符合Hasse原理，即，它们在$\mathbb Q$的每个完成上具有零，但在$\mathbb Q$上没有零。我们还证明了$\mathbb Z$上所有三元三次型的正比例非平凡地满足Hasse原理，即，它们在$\mathbb Q$的每个完成上具有零，并且在$\mathbb Q$上也具有零。类似的结果被证明为其他属一个模型，即，方程的形式$z^2=f(x,y)$ $f$是一个二元四次形式的$\mathbb Z$，和交叉点对二次在$\mathbb P^3$。

同时不满足哈塞原理的图厄方程

Thue equations that simultaneously fail the Hasse principle

改进构造，得到正比例图厄方程，对小于正整数m的h，其F(x; y)=h同时不满足整数哈塞原理，F为固定次数>2的二元形式且按系数绝对值最大者排序。

Two variable polynomial congruences and capacity theory

We use capacity theory to analyze Coppersmith's method for finding small solutions of linear two variable polynomial congruences. We show that the method will succeed in a positive proportion of cases and fail in a different positive proportion of cases. We also bound the number of small solutions via capacity theory.

Q上正比例的三次曲线具有线性行列式表示

A positive proportion of cubic curves over Q admit linear determinantal representations

一个光滑的平面三次体能否由三元线性型元素的方阵的行列式定义？如果可以，我们说它容许线性行列式表示。本文研究了各种域上光滑平面三次函数的线性行列式表示，证明了大域（或充域）上的光滑平面三次函数都有线性行列式表示.由于局部域很大，任何局部域上的光滑平面三次函数都有线性行列式表示。作为应用，我们证明了Q上按高度排序的正比例光滑平面三次函数允许线性行列式表示。我们还证明了，如果Bhargava-Kane-Lenstra-Poonen-Rains关于Selmer群分布的猜想成立，Q上光滑平面三次函数的正比例不符合线性行列式表示存在的局部-全局原理。