Information Theory and Moduli of Riemann Surfaces

TL;DR

This paper explores the application of information theory to reconstruct Riemann surfaces from a small subset of their period matrices, showing that a generic set of moduli can determine the surface with high probability.

Contribution

It introduces a novel approach using information theory to identify moduli of Riemann surfaces from limited period data, demonstrating probabilistic reconstruction.

Findings

Any set of 3g-3 periods can serve as moduli for the surface with high probability.

The period matrix's sparsity allows for efficient reconstruction of the Riemann surface.

Methods from information theory are effective in the context of algebraic geometry.

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.