AI 中文总结
本文给出全局域上K3曲面约化处Picard秩跳跃的精确密度公式，并证明存在无穷多个约化具有固定平方类的Néron-Severi元素，且一般模点对应的密度为零。
AI 中文摘要
设$X$是定义在特征为$0$或$p \geq 3$的全局域$K$上的代数K3曲面，当$K$为全局函数域时假定$X$是普通的。我们给出了$K$的素数$p$的密度的一个精确公式，在这些素数处，约化$X_p$的几何Néron-Severi群的秩由$X$所附的Galois表示给出。作为推广，我们证明了$X$存在无穷多个约化，其几何Néron-Severi群中的元素在相交配对下表示一个固定的平方类，并证明了关于此类位置的密度的结果。特别地，我们证明如果$X$对应于一个足够一般的模点，则满足此条件的$K$的位置的密度为零。
英文摘要
Let $X$ be an algebraic K3 surface defined over a global field $K$ of characteristic $0$ or $p \geq 3$, assumed ordinary in the case where $K$ is a global function field. We provide an exact formula for the density of primes $p$ of $K$ at which the geometric Neron-Severi group of the reduction $X_p$ has a given rank in terms of the Galois representation attached to $X$. As an extension, we prove that there are infinitely many reductions of $X$ with an element of the geometric Neron-Severi group representing a fixed square class under the intersection pairing and prove results about the density of such places. In particular, we show that if $X$ corresponds to a sufficiently generic moduli point, a density zero set of places of $K$ satisfy this condition.
Comments32 pages, comments welcome
