A signed count of torsion points on real abelian varieties

While the number of m-torsion points on an abelian variety of dimension g over the complex numbers is always equal to m^2g, the number of real m-torsion points varies between m^g and m^2g when m is even. I will assign a sign ±1 to each real m-torsion point on a real principally polarized abelian variety such that the sum over all signs is always m^g. I will give an interpretation of this count for Jacobians of curves and for Kummer surfaces. If time permits, I will speculate about generalizations to arbitrary ground fields.