= Solution
The <line bundle translation stabilizer> is the closed subgroup
$$
K(L)=\ker\phi_L=\{x\in X:T_x^*L\cong L\}.
$$
Each difference $x_1-x_2$ lies in $K(L)$, by the equality of <Weil divisors> just proved. Thus the subgroup they generate lies in $K(L)$, and closedness puts its <Zariski topology> closure there too. \b[Therefore $\boxed{Y\subseteq K(L)}$.] The closure of a subgroup is again a subgroup: multiplication and inversion preserve its closure by continuity. In this setting it is connected as well, since $C-C$ is connected and contains zero, and the increasing finite sums of $C-C$ are connected with a common point. Its reduced closure is an abelian subvariety.
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