Algebraic relations among periods and logarithms of rank 2 Drinfeld modules

TL;DR

This paper proves that for rank 2 Drinfeld modules over function fields, the entries of the period matrix are algebraically independent over the base field, leading to new results on the transcendence of Drinfeld logarithms.

Contribution

It establishes the algebraic independence of period matrix entries and Drinfeld logarithms for rank 2 Drinfeld modules, extending transcendence results.

Findings

Transcendence degree of period matrix entries is 4.

Algebraic independence of Drinfeld logarithms of algebraic functions.

Results hold for modules without complex multiplication in odd characteristic.

Abstract

For any rank 2 Drinfeld module rho defined over an algebraic function field, we consider its period matrix P, which is analogous to the period matrix of an elliptic curve defined over a number field. Suppose that the characteristic of F_q is odd and rho is without complex multiplication. We show that the transcendence degree of the field generated by the entries of P over F_q(theta) is 4. As a consequence, we show also the algebraic independence of Drinfeld logarithms of algebraic functions which are linearly independent over F_q(theta).