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Algebraic relations among periods and logarithms of rank 2 Drinfeld modules
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 2, April 2011
- pp. 359-391
- 10.1353/ajm.2011.0009
- Article
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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 ${\Bbb F}_q(\theta)$.