subj: etale covers of line bundles on an abelian variety
Is there an explicit decryption of finite etale covers of a line bundle $L$ on an abelian variety and its associated C*-bundles $L^* L^o = L \setminus A\times {0}$ (i.e. the C*-bundle $L^*$ L^o$ is $L$ without the zero section) ?
Pull-back along multiplication by $n$ map $n:A\rightarrow A$ gives a pull-back $n^*_A L' \rightarrow L$.
A tensor power map $L'\mapsto L'^{\otimes n}$ gives rise to an etale map of
C*-bundles
$L^* L^o \rightarrow L'^{\otimes L^{o\otimes n}$,
and thus if $L$ happens to be a tensor power, to an etale cover of L. $L^o$.
Can we obtain all etale covers of $L^*$ L^o$ this way ?
