In recent work, Abreu, Balazs, Holighaus, Luef and Speckbacher introduced a Gabor transform on the space of periodic Dirac trains which takes a flat torus as its time-frequency plane as well as a corresponding Bargmann type transform. The images of this Bargmann type transform are theta functions on a polarized complex Abelian variety. I plan to discuss the consequences of this viewpoint, briefly for frame results and more extensively for the asymptotic behavior of time-frequency localization operators for Dirac trains with long periods. In particular I want to demonstrate that this viewpoint leads to asymptotic formulas for the trace and spectral counting functions of these operators. Furthermore, one finds that the spectra of operators with indicator functions as their symbol develop "plunge" behaviors.