= Solution
The <Monodromy theorem> states that a germ admitting analytic continuation along every path in a simply connected domain extends to a single-valued <holomorphic function> there. Equivalently continuations along homotopic paths with fixed endpoints agree. For a biholomorphism between the two tori, lift its composition with $\mathbb C\to\mathbb C/\Lambda_1$ through the second covering. Simple connectivity, or monodromy applied to local inverse charts, yields a holomorphic lift $F:\mathbb C\to\mathbb C$.
Lift the inverse similarly. The compositions of the lifts differ from the identity by <lattice> translations; adjusting the lifts shows that $F$ is bijective. By the allowed classification, $F(z)=az+b$ with $a\ne0$. For $\lambda\in\Lambda_1$, its two arguments $z,z+\lambda$ represent the same <torus> point, so $a\lambda=F(z+\lambda)-F(z)\in\Lambda_2$. Thus $a\Lambda_1\subseteq\Lambda_2$. Apply the inverse lift to obtain the reverse inclusion. Therefore
$$
\boxed{\Lambda_2=a\Lambda_1.}
$$
The translation $b$ affects the chosen origin, not the period <lattice>.
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