Algebraic relations among hyperderivatives of periods and logarithms of Drinfeld modules

TL;DR

This paper establishes the algebraic independence of hyperderivatives of periods, quasi-periods, logarithms, and quasi-logarithms of Drinfeld modules, revealing deep structural relations in function field arithmetic.

Contribution

It provides a complete description of algebraic relations among hyperderivatives of key invariants of Drinfeld modules, extending previous results on their algebraic independence.

Findings

All algebraic relations among hyperderivatives are determined.

Hyperderivatives of linearly independent periods and logarithms are algebraically independent.

The results apply to Drinfeld modules over separable closures of rational function fields.

Abstract

We determine all algebraic relations among all hyperderivatives of the periods, quasi-periods, logarithms, and quasi-logarithms of Drinfeld modules defined over a separable closure of the rational function field. In particular, for periods and logarithms that are linearly independent over the endomorphism ring of the Drinfeld module, we prove the algebraic independence of their hyperderivatives and the hyperderivatives of the corresponding quasi-periods and quasi-logarithms.