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arXiv · 1309.2304

Information Theory and Moduli of Riemann Surfaces

James S. Wolper
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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.

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James S. Wolper. 2013-09-09. Information Theory and Moduli of Riemann Surfaces. https://arxiv.org/abs/1309.2304

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