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Abstract

Let $K$ be a number field, $X$ be a smooth projective curve over it and $D$ be a reduced divisor on $X$. Let $(E,\nabla)$ be a vector bundle with connection having meromorphic singularities on $D$. Let $p_1,\dots,p_s\in X(K)$ and $X^o:=\overline X\setminus\{D,p_1,\dots, p_s\}$ (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 arithmetic type 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–Shidlovskiĭ theory. We prove that the value of an $E$-section of arithmetic type at an algebraic point different from the $p_j$'s has maximal transcendence degree. The Siegel–Shidlovskiĭ theorem is a special case of our theorem proved. We give two applications of the theorem.