Horizontal sections of connections on curves and transcendence

C. Gasbarri. "Horizontal sections of connections on curves and transcendence." Acta Arithmetica 158.2 (2013): 99-128. <http://eudml.org/doc/279450>.

@article{C2013,
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,∇) 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.},
author = {C. Gasbarri},
journal = {Acta Arithmetica},
keywords = {transcendence theory; connections; Nevanlinna theory; Siegel–Shidlovskiĭ theorem},
language = {eng},
number = {2},
pages = {99-128},
title = {Horizontal sections of connections on curves and transcendence},
url = {http://eudml.org/doc/279450},
volume = {158},
year = {2013},
}

TY - JOUR
AU - C. Gasbarri
TI - Horizontal sections of connections on curves and transcendence
JO - Acta Arithmetica
PY - 2013
VL - 158
IS - 2
SP - 99
EP - 128
AB - 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.
LA - eng
KW - transcendence theory; connections; Nevanlinna theory; Siegel–Shidlovskiĭ theorem
UR - http://eudml.org/doc/279450
ER -