Let $A$ be an abelian variety over $\mathbb{C}$ and let $X\_m$ the subset of nontrivial $m$-torsion points on $A$. Can we realize $X\_m$ as the zero locus of a global section of a suitable vector bundle $E$ of rank $\dim(A)$ on $A$?


For $\dim(A)=1$ the answer is trivially yes and for $\dim(A)=2$ this should be doable via the Serre construction. What about higher dimensions?



Torsion points of abelian variety as zeros of a section of a vector bundle?
The crucial case is $m=1$: if you have a vector bundle $E$ on $A$ of rank $\dim(A)$ and a section $s$ of $E$ whose zero locus is $\{0\} $, pulling back $(E,s)$ by multiplication by $m$ gives the general case. This question has been studied by O. Debarre, *The diagonal property for abelian varieties*, Contemporary Mathematics 465, AMS (2008), p. 45-50. The answer is positive for Jacobians, but not for general abelian varieties.


