On slopes of isodual lattices

TL;DR

This paper investigates the slope filtration of isodual lattices, a class relevant in various contexts, and provides new cases supporting Bost's conjecture that the tensor product of semistable lattices remains semistable.

Contribution

It studies the slope filtration of isodual lattices and proves new specific cases of Bost's conjecture regarding their semistability under tensor products.

Findings

Identifies properties of isodual lattices relevant to slope filtration.

Proves new particular cases of Bost's conjecture.

Supports the conjecture that tensor products of semistable lattices are semistable.

Abstract

The slope filtration of Euclidean lattices was introduced in works by Stuhler in the late 1970s, extended by Grayson a few years later, as a new tool for reduction theory and its applications to the study of arithmetic groups. Lattices with trivial filtration are called semistable, in keeping with a classical terminology. In 1997, Bost conjectured that the tensor product of semistable lattices should be semistable itself. Our aim in this work is to study these questions for the restricted class of isodual lattices. Such lattices appear in a wide range of contexts, and it is rather natural to study their slope filtration. We exhibit specific properties in this case, which allow, in turn, to prove some new particular cases of Bost's conjecture.