On the classification of lattices over $\Q(\sqrt{-3})$, which are even unimodular $\Z$-lattices

TL;DR

This paper classifies even unimodular lattices over (4) for ranks 4, 8, and 12, constructs their theta series as Hermitian modular forms, and analyzes the cusp form filtration.

Contribution

It provides a complete classification of certain lattices over (4), linking lattice theory with Hermitian modular forms and cusp form analysis.

Findings

Classified lattices of ranks 4, 8, 12 over (4)

Constructed associated theta series as Hermitian modular forms

Computed the filtration of cusp forms

Abstract

We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are even and unimodular \Z-lattices. Using this classification we construct the associated theta series, which are Hermitian modular forms, and compute the filtration of cusp forms.