Horizontal sections of connections on curves and transcendence

TL;DR

This paper generalizes the concept of E-functions to sections of vector bundles with connections on algebraic curves, proving their values at algebraic points have maximal transcendence degree, with applications to isomonodromic connections.

Contribution

It introduces a new notion of E-sections of type alpha for vector bundles with connections on curves, extending classical transcendence results.

Findings

E-sections of type alpha have maximal transcendence degree at algebraic points.

The classical Siegel-Shidlovsky theorem is a special case of the new result.

Application to isomonodromic connections demonstrates the theory's relevance.

Abstract

Let $K$ be a number field, $\UX$ be a smooth projective curve over it and $D$ be a reduced divisor on $\UX$. Let $(E,\nabla)$ be a fibre bundle with connection having meromorphic poles on $D$. Let $p_1,...,p_s\in\UX(K)$ and $X:=\UX\setminus\{D,p_1,..., p_s\}$ (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 type $\alpha$ 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 Shidlowski theory. We prove that the value of a $E$--section of type $\alpha$ in an algebraic point different from the $p_j$'s has maximal transcendence degree. Siegel Shidlowski theorem is a special case of the theorem proved. We give an application to isomonodromic connections.