Algebraic relations among periods and logarithms of rank 2 Drinfeld modules

Abstract

For any rank $2$ Drinfeld module $\rho$ defined over an algebraic function field, we consider its period matrix $P_{\rho}$, which is analogous to the period matrix of an elliptic curve defined over a number field. Suppose that the characteristic of the finite field ${\Bbb F}_q$ is odd and that $\rho$ does not have complex multiplication. We show that the transcendence degree of the field generated by the entries of $P_{\rho}$ over ${\Bbb F}_q(\theta)$ is $4$. As a consequence, we show also the algebraic independence of Drinfeld logarithms of algebraic functions which are linearly independent over ${\Bbb F}_q(\theta)$.