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 and (the ’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 ; 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 ’s has maximal transcendence degree. The Siegel-Shidlovskiĭ theorem is a special case of our theorem proved. We give two applications of the theorem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-2-1,
author = {C. Gasbarri},
title = {Horizontal sections of connections on curves and transcendence},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {99-128},
zbl = {1288.11072},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-2-1}
}
C. Gasbarri. Horizontal sections of connections on curves and transcendence. Acta Arithmetica, Tome 161 (2013) pp. 99-128. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-2-1/