A Mordell Inequality for Lattices over Maximal Orders

TL;DR

This paper establishes a Mordell-type inequality for lattices over maximal orders in complex and quaternionic Hermitian spaces, demonstrating the optimality of the Barnes-Wall lattice in 16 dimensions with Hurwitz structures.

Contribution

It introduces a new inequality for lattices over maximal orders, extending Mordell's inequality to complex and quaternionic settings, and proves the Barnes-Wall lattice's optimal density.

Findings

Barnes-Wall lattice is optimally dense among Hurwitz-structured lattices in 16 dimensions

Established a Mordell inequality analogue for lattices over maximal orders

Extended classical inequalities to complex and quaternionic Hermitian spaces

Abstract

In this paper we prove an analogue of Mordell's inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that the 16-dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz structures.