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Horizontal sections of connections on curves and transcendence

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Gasbarri C.

Acta Arithmetica

2013

tom 158

nr 2

99-128

Let K be a number field, X be a smooth projective curve over it and D be a reduced divisor on X. Let (E,∇) be a vector bundle with connection having meromorphic singularities on D. Let $p_1,...,p_s ∈ X(K)$ and $X^o:=X̅∖{D,p_1,..., p_s}$ (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 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.

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1 Acta Arithmetica

1 Gasbarri C.

1 2013

Wyniki wyszukiwania

Horizontal sections of connections on curves and transcendence