= Solution
Let $V=H^0(C,\Omega_C^1)$, of complex dimension $g$, and choose a <symplectic basis> $a_1,\ldots,a_g,b_1,\ldots,b_g$ of $H_1(C,\mathbb Z)$, with $a_i\cdot b_j=\delta_{ij}$. Integration gives a homomorphism
$$
I:H_1(C,\mathbb Z)\longrightarrow V^\vee,\qquad I(\gamma)(\omega)=\int_\gamma\omega.
$$
We first prove both injectivity and discreteness, rather than treating periods as a lattice by definition.
Cut the <compact Riemann surface> along this basis to a polygon. For closed one-forms $\omega,\eta$, choose a primitive $F$ of $\omega$ on the cut polygon and apply <Stokes theorem> to $d(F\eta)=\omega\wedge\eta$. Pairing its boundary edges records the jumps of $F$ across the cuts and gives the <Riemann bilinear relations for a compact surface>:
$$
\int_C\omega\wedge\eta=\sum_i\left(A_i(\omega)B_i(\eta)-B_i(\omega)A_i(\eta)\right),\quad A_i(\omega)=\int_{a_i}\omega,\ B_i(\omega)=\int_{b_i}\omega.
$$
If a <holomorphic differential form> $\omega$ has all $a$-periods zero, take $\eta=\overline\omega$. The right side is zero, while in a holomorphic coordinate, for $\omega=f(z)\,dz$,
$$
\frac{i}{2}\omega\wedge\overline\omega=|f(z)|^2\,dx\wedge dy.
$$
Its integral is strictly positive unless $\omega=0$. Thus the $a$-period map $V\to\mathbb C^g$ is injective, hence an isomorphism. Choose a basis $\omega_1,\ldots,\omega_g$ with $\int_{a_j}\omega_i=\delta_{ij}$, and put $\tau_{ij}=\int_{b_j}\omega_i$. Applying the same boundary identity to $\omega_i,\omega_j$, whose wedge product is zero, gives $\tau_{ij}=\tau_{ji}$. Applying it to $\omega=\sum c_i\omega_i$ and its conjugate gives
$$
\frac{i}{2}\int_C\omega\wedge\overline\omega=c^{\mathsf T}(\operatorname{Im}\tau)\overline c>0\qquad(c\ne0).
$$
Hence $\operatorname{Im}\tau$ is positive definite.
In the dual coordinates, $I(\sum m_ia_i+n_ib_i)=m+\tau n$. The real-linear map $(x,y)\mapsto x+\tau y$ from $\mathbb R^{2g}$ to $\mathbb C^g$ is an isomorphism: taking imaginary parts of a zero gives $(\operatorname{Im}\tau)y=0$, then $y=0$ and $x=0$. Its image of $\mathbb Z^{2g}$ is therefore discrete and cocompact, and no nonzero integral cycle has zero image. Thus
$$
\boxed{I\text{ is injective, with full lattice }\Lambda=\mathbb Z^g+\tau\mathbb Z^g\subset V^\vee.}
$$
The <Abel-Jacobi map of a compact Riemann surface> sends an effective <complex analytic divisor> $D=\sum Q_i$ to $\sum_i\int_{P_0}^{Q_i}(\omega_1,\ldots,\omega_g)$ modulo $\Lambda$. Changing a path adds an integral period, so the map is well-defined. For two degree-$d$ <complex analytic divisors>, let $E=D-D'=\sum_P n_PP$, with $\sum n_P=0$, and put
$$
v_i=\sum_P n_P\int_{P_0}^P\omega_i.
$$
We prove that $E$ is principal exactly when $v\in\Lambda$, including the construction of a meromorphic function.
First construct a <differential of the third kind> $\eta$ with residues $n_P$ at the support of $E$ and no other poles. For the reduced support $S$, the residue exact sequence is
$$
0\longrightarrow K_C\longrightarrow K_C(S)\longrightarrow\bigoplus_{P\in S}\mathbb C_P\longrightarrow0.
$$
On <sheaf cohomology>, the obstruction to prescribing residues belongs to $H^1(C,K_C)\cong\mathbb C$ by <Serre duality>; its pairing with the constant section of $\mathcal O_C$ is the sum of residues. This follows directly by integrating the local simple-pole terms around the boundary circles. Thus the obstruction is zero because $\sum n_P=0$, and the required $\eta$ exists. Subtract a linear combination of the normalized <holomorphic differential forms> so that all its $a$-periods are zero.
Take its poles and the integration paths inside the same cut polygon. Apply the boundary calculation above to a primitive $F_i(P)=\int_{P_0}^P\omega_i$, removing small circles around the poles of $\eta$. Since $d(F_i\eta)=0$ away from the poles, the paired outer edges contribute $B_i(\eta)$, while the small circles contribute $2\pi i\sum_P n_PF_i(P)$ by the <residue theorem>. Therefore
$$
\int_{b_i}\eta=2\pi i\,v_i.
$$
The same argument without normalizing $\eta$ gives the useful reciprocity identity
$$
B_i(\eta)-\sum_j\tau_{ij}A_j(\eta)=2\pi i\,v_i.
$$
These identities use compatible cuts and paths; the final conclusions are unchanged by integral periods.
Suppose $v=m+\tau n$ with $m,n\in\mathbb Z^g$. Define
$$
\eta'=\eta-2\pi i\sum_j n_j\omega_j.
$$
Its $a$-periods are $-2\pi i n_i$, and its $b$-periods are $2\pi i m_i$. Around a pole its integral is $2\pi i n_P$. These cycles generate the first <homology group> of the punctured surface, so every period of $\eta'$ belongs to $2\pi i\mathbb Z$. Choosing an auxiliary point $Q$ away from the poles, set
$$
f(P)=\exp\left(\int_Q^P\eta'\right).
$$
It is single-valued away from the poles. Locally $\eta'=n_P\,dt/t+\text{a holomorphic one-form}$, so $f=t^{n_P}$ times a nonvanishing holomorphic function. It extends meromorphically with $\operatorname{div}(f)=E$. This proves sufficiency and explicitly produces the required principal <complex analytic divisor>.
Conversely, if $E=\operatorname{div}(f)$, the meromorphic differential $df/f$ has residues $n_P$ and every period in $2\pi i\mathbb Z$, since each integral counts the winding number of $f$ about zero. Write its $a$-period vector as $2\pi i a$ and its $b$-period vector as $2\pi i b$, with $a,b\in\mathbb Z^g$. The reciprocity identity gives $v=b-\tau a\in\Lambda$. Since $E$ has degree zero, changing the base point cancels in its Abel sum; a base point at the support of $E$ causes no problem, because the Abel integrals use holomorphic forms. We have proved
$$
\boxed{\operatorname{AJ}(D)=\operatorname{AJ}(D')\iff D-D'\text{ is principal}\iff D\sim D'.}
$$
This establishes the <Abel theorem for divisors> directly from the period and residue calculations.
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