$NS(T)\cong\Lambda^\wedge=Hom_{Ab}(\Lambda,C^\star)$ where $T=\frac{C^g}{\Lambda}$ is an abelian variety and $NS(T)$ is defined in terms of $\frac{\{\text{algebraically equivalent to $0$ divisors}\}}{\{\text{linearly equivalent to $0$ divisors}\}}$. Divisors algebraic equivalent to $0$ divisors are divisors having $0$/trivial alternating hermitian form.
$\textbf{Q1:}$ Can I define characters $f\in Hom_{Ab}(\Lambda,C^\star)$'s $log(f)$? I could lift the map to $C$ by $C$ being $C^\star$ universal covering under exponential map if $f$'s domain is path connected. However, $\Lambda$ is a discrete subgroup. What does it mean to lift $f$ here? If there is such $f_1=log(f)$, then $f_1(\lambda+\lambda')-f_1(\lambda)-f_1(\lambda)=0(mod Z)$.
$\textbf{Q2:}$ Suppose I can define the lift $\tilde{f}$ as in $Q1$. Since $T$ is abelian variety, it has a non-degenerate positive divisor $a$ which has associated $\theta$ function. Suppose $\forall \lambda\in\Lambda, \theta(z+\lambda)=\theta(z)\exp(2\pi i(L(z,\lambda)+F(\lambda))$ where $L$ is $C-$linear in $z$ and $R-$bilinear in both slots and $\forall \lambda,\lambda'\in\Lambda, F(\lambda+\lambda')-F(\lambda)-F(\lambda')=L(\lambda,\lambda')(mod Z)$. Then by existence of $\theta$ function given $L$ and $F$, I can modify $F$ by $\frac{log(f)}{2\pi i}$ as above, I can obtain a modified $\theta_1$ function. Then $\psi=\frac{\theta_1}{\theta}$ has property $\psi(z+\lambda)=\psi(z)f(\lambda)$. Is this correct reasoning as the book says every character $f\in\Lambda^\wedge$ comes from some quotient $\theta$ functions by $T$ being abelian variety?
The punchline is to show $NS(T)\to Hom_{Ab}(\Lambda,C^\star)$ is surjective map by taking the automorphy factors out.
Ref. Sec 8. Analytic Theory of Abelian Varieties by Swinnerton-Dyer