Information Theory and Moduli of Riemann Surfaces

Abstract

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.