Horizontal sections of connections on curves and transcendence

C. Gasbarri¹

¹ Université de Strasbourg IRMA 7 rue René Descartes 67084 Strasbourg, France

Acta Arithmetica, Tome 158 (2013) no. 2, pp. 99-128.

Résumé

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.

Détail

DOI : 10.4064/aa158-2-1

Keywords: number field smooth projective curve reduced divisor nabla vector bundle connection having meromorphic singularities dots overline setminus dots may support using tools nevanlinna theory formal geometry definition e section arithmetic type vector bundle respect points natural generalization notion e function defined siegel shidlovski theory prove value e section arithmetic type algebraic point different has maximal transcendence degree siegel shidlovski theorem special theorem proved applications theorem