Let $\Lambda \subset V=\mathbb{C}^g$ be a lattice and $T_g=V/\Lambda$ be a $g$ dimensional complex torus.
According to the Appell-Humbert theorem, any line bundle $L$ of $T_g$ is isomorphic to the line bundle $L(H,\alpha)$ arising from a Hermitian form $H$ whose imaginary part $\mathrm{Im}\ H$ is integer valued on $\Lambda$ and a semi character $\alpha:\Lambda\to\mathbb{C}^1$ (i.e. $\alpha(u+v)=e^{i\pi\operatorname{Im}H(u,v)}\alpha(u)\alpha(v)$).
($\mathbb{C}^1$ is the set of complex numbers its absolute value $=1$ )
More precise, $L(H,\alpha):=\Lambda\setminus (V\times\mathbb{C})$ where $\Lambda \curvearrowright (V\times\mathbb{C})$ is given by $u.(x,z):=(x+u,e_u(x)z)\ (e_u(x):=\alpha(u)e^{\pi H(x,u)+\pi H(u,u)/2})$.
Since the automorphy factors $e_u(x)$ satisfies "1-cocycle condition" (this means $e_{u'+u}(x)=e_{u'}(x+u)e_u(x)$) , this is certainly a group action.
Question. I want to check $\pi:L(H,\alpha)\to T_g;[(x,z)]\mapsto x \pmod{\Lambda}$ is certainly a line bundle along the definition. But I can't find its local trivializations $\{U_i\}_i,\{\tau_i:U_i\times \mathbb{C}\to \pi^{-1}(U_i)\}_i$ explicitly.
In addition, I am also confused that "1-cocycle condition" seems to be different from the ordinary 1-cocycle condition. I learned at differential geometry class, 1-cocycle condition for transition functions $\{g_{ij}:U_i\cap U_j\to GL_r(\mathbb{C})\}_{ij}$ is $g_{jk}(p)g_{ij}(p)=g_{ik}(p)\ (p\in U_i\cap U_j\cap U_k)$.