Information Theory and Moduli of Riemann Surfaces

One interpretation of Torelli’s Theorem, which asserts that a compact Riemann Surface X of genus g > 1 is determined by the g(g+1)/2 entries of the period matrix, is that the period matrix is a message about X. Since this message depends on only 3g− 3 moduli, it is sparse, or at least approximately so, in the sense of information theory. Thus, methods from information theory may be useful in reconstructing the period matrix, and hence the Riemann surface, from a small subset of the periods. The results here show that, with high probability, any set of 3g − 3 periods form moduli for the surface.