= Solution
First record the unlettered definitions. A <divisor on an algebraic curve> on a <compact> <Riemann surface> is a finite formal sum $D=\sum_p n_p[p]$ with $n_p\in\mathbb Z$. Its degree is $\deg D=\sum_p n_p$. A <principal divisor> is $(f)=\sum_p\operatorname{ord}_p(f)[p]$ for a nonzero <meromorphic function>, with positive order at a zero and negative order at a pole. Two divisors are <linearly equivalent> if their difference is principal. Principal divisors form a subgroup because $(fg)=(f)+(g)$ and $(1/f)=-(f)$.
For the doubly periodic function, put $h=f'/f$. It is periodic with the same <period lattice>, and its <residue> at $p_i$ equals $n_i$. Opposite sides of the <fundamental parallelogram> are translates by a period with opposite directions, so their integrals of $h$ cancel. The <residue theorem> gives
$$
0=\oint_{\Gamma_a}h(w)\,dw=2\pi i\sum_i n_i.
$$
Thus
$$
\boxed{\sum_i n_i=0.}
$$
In particular an <elliptic function> has equally many zeros and poles counted with multiplicity in a period cell.
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