Planes in $\mathbb{Z^4}$ and Eisenstein series

TL;DR

This paper investigates the enumeration of 2D sublattices within four-dimensional integer lattices and links the counting problem to Eisenstein series on algebraic groups, revealing deep connections between lattice geometry and automorphic forms.

Contribution

It introduces a Dirichlet series associated with counting sublattices and establishes its relation to Eisenstein series on $GL_4$ and its metaplectic cover, bridging lattice enumeration and automorphic forms.

Findings

Derived a Dirichlet series for counting sublattices

Connected the series to Eisenstein series on $GL_4$

Explored the relation to automorphic forms and lattice geometry

Abstract

We study the number of two-dimensional sublattices of $\mathbb{Z^4}$ of a fixed covolume and construct the associated Dirichlet series. The latter is shown to be related to Eisenstein series on both $GL_4$ and its metaplectic double cover.