Periods of third kind for rank 2 Drinfeld modules and algebraic independence of logarithms

TL;DR

This paper introduces a new notion of periods of third kind for rank 2 Drinfeld modules over function fields, derives explicit formulas, and proves algebraic independence of certain logarithms, advancing understanding of their algebraic relations.

Contribution

It defines periods of third kind for rank 2 Drinfeld modules, provides explicit formulas, and establishes algebraic independence results for logarithms in the CM case.

Findings

Explicit formulas for periods of third kind for rank 2 Drinfeld modules.

Proof of algebraic independence of rho-logarithms of algebraic points.

Complete characterization of algebraic relations among periods of various kinds in odd characteristic.

Abstract

In analogy with the periods of abelian integrals of differentials of third kind for an elliptic curve defined over a number field, we introduce a notion of periods of third kind for a rank 2 Drinfeld Fq[t]-module rho defined over an algebraic function field and derive explicit formulae for them. When rho has complex multiplication by a separable extension, we prove the algebraic independence of rho-logarithms of algebraic points that are linearly independent over the CM field of rho. Together with the main result in [CP08], we completely determine all the algebraic relations among the periods of first, second and third kinds for rank 2 Drinfeld Fq[t]-modules in odd characteristic.