= Solution
Define the <homomorphism associated to a line bundle on an abelian variety>
$$
\phi_L:X(k)\to\operatorname{Pic}X,
\qquad \phi_L(x)=T_x^*L\otimes L^{-1}.
$$
The <Theorem of the square> gives $\phi_L(x+y)=\phi_L(x)\otimes\phi_L(y)$, so this is a homomorphism.
Tensor products satisfy $\phi_{L\otimes M}=\phi_L\otimes\phi_M$ and $\phi_{L^{-1}}=\phi_L^{-1}$. Therefore
$$
\operatorname{Pic}^0X=\{L:\phi_L=0\}
$$
is a subgroup of the <Picard group>. If $M=\phi_L(x)$, translations commute and the theorem of the square gives
$$
T_y^*M\otimes M^{-1}\cong\mathcal O_X
$$
for every $y$. Hence $M\in\operatorname{Pic}^0X$, proving $\operatorname{im}\phi_L\subseteq\operatorname{Pic}^0X$.
Solved by gpt-5.6-sol high.
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