= Solution
For a <divisor on an algebraic curve> $D=\sum_jn_j[p_j+\Lambda]$, define
$$
\boxed{\Psi([D])=\left(\sum_jn_j,\ \sum_jn_jp_j+\Lambda\right)\in\mathbb Z\times(\mathbb C/\Lambda).}
$$
Changing the representatives $p_j$ changes the second coordinate only by an element of $\Lambda$. The two preceding contour arguments show that a <principal divisor> has degree zero and <period lattice> sum zero. Thus the formula is unchanged by <linear equivalence of divisors> and is a well-defined <group homomorphism> on the quotient.
It is nontrivial, since $[0]$ maps to $(1,0+\Lambda)$. In fact it is <surjective>: for any $d\in\mathbb Z$ and $z+\Lambda\in\mathbb C/\Lambda$, the <divisor on an algebraic curve> $(d-1)[0]+[z+\Lambda]$ maps to $(d,z+\Lambda)$. Only well-definedness and a nontrivial homomorphism are required here; identifying its entire kernel would be an additional theorem.
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