Line bundle translation stabilizer ($K(L)=\ker {\varphi }_L$)

For a line bundle $L$ on an abelian variety $A$, this closed subgroup consists of $a$ such that $T_a^{\ast }L\cong L$. It is a stabilizer of a Picard group class, which need not be the same as the stabilizer of an individual Weil divisor representing that class.