Torsion points on cohomology support loci: from D-modules to Simpson's theorem

We study cohomology support loci of regular holonomic D-modules on complex abelian varieties, and obtain conditions under which each irreducible component of such a locus contains a torsion point. One case is that both the D-module and the corresponding perverse sheaf are defined over a number field; another case is that the D-module underlies a grade-polarizable mixed Hodge module with a Z-structure. As a consequence, we obtain a new proof for Simpson's result that Green-Lazarsfeld sets are translates of subtori by torsion points.