有限域上有理函数的等价类数量

Rational points on twisted K3 surfaces and derived equivalences

Using a construction of Hassett--Várilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel.

Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

We consider the reduction of Brauer classes on surfaces over number fields, with a view toward applications to rationality and derived equivalence. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial upon reduction for a set of places of positive natural density. As a consequence, there are cubic fourfolds which become rational upon reduction for a positive proportion of places, and there are twisted derived equivalent K3 surfaces which become derived equivalent upon reduction for a positive proportion of places.

Rational points and derived equivalence

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a pair of hyperkaehler 4-folds over Q. The latter is independently interesting as a new example of a transcendental Brauer-Manin obstruction to the Hasse principle.

三次有理函数在有限域上的分类

Classification of Rational Functions of Degree Three over Finite Fields

本文研究了有限域上三次有理函数在PGL等价下的分类问题，通过分析值频率和分支点，给出了在奇特征下三次有理函数的等价类数量公式，完成了分类工作。

The converse of Bohr's equivalence theorem with Fourier exponents linearly independent over the rational numbers

Given two arbitrary almost periodic functions with associated Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip $V$, where both functions assume the same set of values on every open vertical substrip included in $V$, is a necessary and sufficient condition for both functions to have the same region of almost periodicity and to be $^*$-equivalent or Bohr-equivalent. This result represents the converse of Bohr's equivalence theorem for this particular case.

Equivalence classes of dessins d'enfants with two vertices

Let $N$ be a positive integer. For any positive integer $L\leq N$ and any positive divisor $r$ of $N$, we enumerate the equivalence classes of dessins d'enfants with $N$ edges, $L$ faces and two vertices whose automorphism groups are cyclic of order $r$. Further, for any non-negative integer $h$, we enumerate the equivalence classes of dessins with $N$ edges, $h$ faces of degree $2$ with $h\leq N$, and two vertices, whose automorphism groups are cyclic of order $r$. Our arguments are essentially based upon a natural one-to-one correspondence of the equivalence classes of all dessins with $N$ e…

Rational Equivalences on Products of Elliptic Curves in a Family

Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map $\text{CH}^1(E_1)_0\otimes\text{CH}^1(E_2)_0\to\text{CH}^2(E_1\times E_2)$, and a conjecture due to Beilinson predicts that the image of this map is finite when $k$ is a number field. We construct a $2$-parameter family of elliptic curves that can be used to produce examples of pairs $E_1,E_2$ where this image is finite. The family is constructed to guarantee the existence of a rational curve passing through a specified point in the Kummer surface of $E_1\times E_2$.

有理Tate类

Rational Tate classes

正如德利涅（Deligne，2000）所说，在证明霍奇和泰特定理的绝望中，我们可以尝试寻找替代品。对于特征为零的阿贝尔簇，Deligne（1982）构建了一个霍奇类的理论，该理论具有许多性质，如果霍奇猜想已知，代数类将具有这些性质。在这篇文章中，我调查是否存在一个理论的“理性泰特类”品种在有限域的属性，代数类将有如果霍奇和泰特代数是已知的。v3.提交的版本。