= Solution
Since $X_j\sim\operatorname{Poisson}(n_j\lambda_j)$, the rate MLE has asymptotic variance $\lambda_j/n_j$. The <delta method> for the logarithm gives
$$
\operatorname{Var}(\log\widehat\lambda_j)\simeq\frac1{n_j\lambda_j}.
$$
Independence of the arms therefore yields
$$
Z=\frac{\log\widehat\lambda_T-\log\widehat\lambda_C}
{\sqrt{1/(n_T\widehat\lambda_T)+1/(n_C\widehat\lambda_C)}}.
$$
Under $H_0:\lambda_T=\lambda_C=\lambda$, the asymptotic variance of the log rate ratio is
$$
\frac1{n_T\lambda}+\frac1{n_C\lambda}.
$$
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