Prolongations of t-motives and algebraic independence of periods

TL;DR

This paper proves algebraic independence of certain periods related to t-motives, introduces a prolongation construction for t-motives, and establishes hypertranscendence of the Anderson-Thakur function and its derivatives.

Contribution

It introduces the prolongation construction for t-motives and proves algebraic independence of period lattice generators for tensor powers of the Carlitz module.

Findings

Coordinates of period lattice generators are algebraically independent when n is coprime to the characteristic.

The prolongation construction provides a framework for understanding algebraic independence in t-motives.

The Anderson-Thakur function and its hyperderivatives are algebraically independent, demonstrating hypertranscendence.

Abstract

In this article we show that the coordinates of a period lattice generator of the $n$-th tensor power of the Carlitz module are algebraically independent, if $n$ is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for $t$-motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another ingredient is a theorem which shows hypertranscendence for the Anderson-Thakur function $\omega(t)$, i.e. that $\omega(t)$ and all its hyperderivatives with respect to $t$ are algebraically independent.