标题： 关于$p$-进解到KZ方程、普通晶体和$p^s$-超几何解 显示英文标题

标题： On $p$-adic solutions to KZ equations, ordinary crystals, and $p^s$-hypergeometric solutions

摘要： We consider the KZ connection associated with a family of hyperelliptic curves of genus $g$ over the ring of $p$-adic integers $\mathbb{Z}_p$. Then the dual connection is the Gauss-Manin connection of that family. We observe that the Gauss-Manin connection has an ordinary $F$-crystal structure and its unit root subcrystal is of rank $g$. We prove that all local flat sections of the KZ connection annihilate the unite root subcrystal, and the space of all local flat sections of the KZ connection is a free $\mathbb{Z}_p$-module of rank $g$. 我们还考虑了任意$s\geq 1$的单位根子晶格对$p^s$的约化。 我们证明其零化子由所谓的$p^s$-超几何平坦截面生成。 特别是，这意味着在$p^s$下的 KZ 连接在$\mathbb{Z}_p$上的任意局部平坦截面的约化是$p^s$-超几何平坦截面的线性组合。 显示英文摘要

摘要： We consider the KZ connection associated with a family of hyperelliptic curves of genus $g$ over the ring of $p$-adic integers $\mathbb{Z}_p$. Then the dual connection is the Gauss-Manin connection of that family. We observe that the Gauss-Manin connection has an ordinary $F$-crystal structure and its unit root subcrystal is of rank $g$. We prove that all local flat sections of the KZ connection annihilate the unite root subcrystal, and the space of all local flat sections of the KZ connection is a free $\mathbb{Z}_p$-module of rank $g$. We also consider the reduction modulo $p^s$ of the unit root subcrystal for any $s\geq 1$. We prove that its annihilator is generated by the so-called $p^s$-hypergeometric flat sections of the KZ connection. In particular, that means that the reduction modulo $p^s$ of an arbitrary local flat section of the KZ connection over $\mathbb{Z}_p$ is a linear combination of the $p^s$-hypergeometric flat sections.