  In this paper we point out the natural relation between $\mathbb Q$-twisted
objects of the derived category of abelian varieties, cohomological rank
functions, and semihomogeneous vector bundles. We apply this to two basic
classes of corresponding objects via the Fourier-Mukai-Poincar\'e transform:
positive twists of the ideal sheaf of one point, and evaluation complexes of
ample semihomogeneous vector bundles. This leads to $\mathbb Q^{\ge 0}$- graded
section modules associated to line bundles on abelian varieties (containing the
usual section rings), built by means of semihomogeneous vector bundles. We
prove a duality relation between such section modules associated to dual
polarizations. Other applications include formulas relating the thresholds of
relevant cohomological rank functions appearing in this context. As a
consequence we produce non-trivial obstructions to surjectivity of
multiplication maps of global sections of certain line bundles.
