曲线上联络的水平截面与超越
Horizontal sections of connections on curves and transcendence
摘要：
设$K$是数域，$\UX$是其上的光滑射影曲线，$D$是$\UX$上的约化因子.设$(E,\nabla)$是$D$上具有亚纯极点联络的纤维丛.令$p_1,...,p_s\in\UX(K)$和$X:=\UX\setminus\{D,p_1,..., p_s\}$（$p_j$可以支持$D$）。利用Nevanlinna理论和形式几何的工具，给出了向量丛$E$关于点$p_j$的$α$型截面$E$的定义，这是Siegel Shidlowski理论中定义的$E$函数概念的自然推广.证明了$E$ --型$α$--截面在不同于$p_j$的代数点上的值具有最大超越度. Siegel Shidlowski定理是该定理的一个特例证明。我们给出了等单径联络的一个应用。
英文摘要：
Let $K$ be a number field, $\UX$ be a smooth projective curve over it and $D$ be a reduced divisor on $\UX$. Let $(E,\nabla)$ be a fibre bundle with connection having meromorphic poles on $D$. Let $p_1,...,p_s\in\UX(K)$ and $X:=\UX\setminus\{D,p_1,..., p_s\}$ (the $p_j$'s may be in the support of $D$). Using tools from Nevanlinna theory and formal geometry, we give the definition of $E$--section of type $α$ of the vector bundle $E$ with respect to the points $p_j$; this is the natural generalization of the notion of $E$ function defined in Siegel Shidlowski theory. We prove that the value of a $E$--section of type $α$ in an algebraic point different from the $p_j$'s has maximal transcendence degree. Siegel Shidlowski theorem is a special case of the theorem proved. We give an application to isomonodromic connections.
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