Sparse Modular Forms, Lattices, and Codes

TL;DR

This paper explores the sparseness properties of modular forms, lattices, and codes, especially in the context of holographic CFTs, by analyzing their behavior as key parameters grow large.

Contribution

It introduces a sparseness condition for lattices derived from codes and constructs examples of sparse modular forms and lattices, providing new insights into their asymptotic behavior.

Findings

Constructed sparse families of modular forms like Eisenstein series

Defined sparseness conditions for lattices from codes

Provided evidence that lattices from Reed-Muller codes are sparse

Abstract

Motivated by sparseness conditions for holographic CFTs, we investigate sparseness of modular forms, lattices, and codes. For this we investigate the free energy of such objects as their weight, dimension or size goes to infinity. We construct families of modular forms that are sparse, such as the Eisenstein series $E_{2k}(\tau)$. We then investigate lattices that come from codes and introduce a sparseness condition for such lattices. We investigate the limit of lattices constructed from self-dual Reed-Muller codes and provide evidence that they are sparse in this sense.