Transcendence of analytic parameters of rational elliptic modules

The author considers rational elliptic modules over an algebraically closed complete extension $K\supset\mathbf F_q(T)$ and proves that $e(1)$ is transcendental over $\mathbf F_q(T)$ for modules of rank $m<q$. Transcendence of the periods of the lattices corresponding to rational modules of the form $\varphi(t)=T\mathscr F^0+a\mathscr F^m$ is also proved.
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