The Schottky Problem is to characterize the Jacobians in the spaceof principally polarized abelian varieties (PPAVs). Besides its intrinsic interest, it is related to deep questions in PDE and Physics. The perspective of Information Theory leads to the numerical investigation of properties of the distribution of the periods, especially in cases of relatively large genus. In turn, numerical results lead to conjectures that the periods of hyperelliptic curves are band-limited, and that the squared moduli of the periods of a general surface are Zipfian.