Pseudo-normalized Hecke eigenform and its application to extremal $2$-modular lattices

TL;DR

This paper introduces the concept of pseudo-normalized Hecke eigenforms and demonstrates their application in proving that extremal 2-modular lattices of ranks 32 and 48 are generated by their minimal vectors.

Contribution

The paper introduces the concept of pseudo-normalized Hecke eigenforms and applies it to analyze the structure of extremal 2-modular lattices.

Findings

Extremal 2-modular lattices of ranks 32 and 48 are generated by their minimal vectors.

Properties of the difference of normalized Hecke eigenforms are utilized in the proof.

Introduction of pseudo-normalized Hecke eigenforms as a new concept in lattice theory.

Abstract

It is shown that extremal $2$-modular lattices of ranks $32$ and $48$ are generated by their vectors of minimal norm. In the proof, we use certain properties of the difference of normalized Hecke eigenforms. We refer to them as the pseudo-normalized Hecke eigenform, the concept of which is introduced in this paper.