普通K3曲面族的S-整性及代数性定理
S-integrality for families of ordinary K3 surfaces and algebraicity theorems
AI总结：
本文证明正特征GSpin Shimura簇上特殊除子的S-整性定理，通过形式特殊自同态的代数性定理结合丢番图逼近技术，并利用点态单演定理及Mumford-Tate猜想的正特征类比完成证明。
AI中文摘要：
我们证明了正特征下GSpin Shimura簇上特殊除子的S-整性定理。设C是这样一个Shimura簇中的一条一般普通曲线，且不包含在任何特殊除子中。那么，对于任意递增的与p互素的正整数序列m_i以及C上任意有限闭点集S⊂C，我们证明C\S对所有但有限多个i都与特殊除子Z(m_i)相交。关键的新输入是关于形式特殊自同态的代数性定理，它使我们能够使用丢番图逼近的技术。我们利用点态单演定理和Mumford-Tate猜想的正特征类比来证明这个代数性定理。
英文摘要：
We prove an $S$-integrality theorem for special divisors on GSpin Shimura varieties in positive characteristic. Let $C$ be a generically ordinary curve in such a Shimura variety not contained in any special divisor. Then, for any increasing sequence of prime-to-$p$ positive integers $m_i$ and any finite set of closed points $S\subset C$, we prove that $C\setminus S$ meets the special divisor $Z(m_i)$ for all but finitely many $i$. The key new input is an algebraicity theorem for formal special endomorphisms which allows us to use techniques from Diophantine approximation. We prove this algebraicity theorem using a punctual monodromy theorem and a positive-characteristic analogue of the Mumford--Tate conjecture.
