论文标题
Manin-Mumford在属2中的猜想和K3表面上的理性曲线
The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces
论文作者
论文摘要
让$ a $成为一个简单的Abelian表面，这是代数封闭的$ k $。让$ s \ subset a（k）$是$ a $ $ a $的扭转点$ x $ x $ a $，以便存在$ 2 $ curve $ c $和a map $ f：c \ to $ x $，以至于$ x $属于$ f $ of $ f $，$ f $ sends a weierstrass of $ c $ $ c $ a $ c $ oild $ a $ a $ a $ a $。本说明的目的是表明，如果$ k $具有特征性零，则$ s $是有限的 - 这与$ k $是有限场的代数关闭的情况形成对比，其中$ s = a（k）$，如Bogomolov和Tschinkel所示。我们推断出，如果$ k = \ bar {\ mathbb {q}} $，与$ a $相关的kummer表面具有无限的许多$ k $ - 未包含在$ a $ $ a $的属$ 2 $曲线中的理性曲线中，则与代数封闭的情况相反。
Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surface associated to $A$ has infinitely many $k$-points not contained in a rational curve arising from a genus $2$ curve in $A$, again in contrast to the situation over the algebraic closure of a finite field.
