Lattices from Hermitian function fields

TL;DR

This paper studies lattices derived from Hermitian function fields, showing they are generated by minimal vectors, estimating their number, and analyzing their automorphism groups, advancing understanding of algebraic function field lattices.

Contribution

It introduces new properties of Hermitian function field lattices, including generation by minimal vectors and automorphism group characteristics, expanding prior work on elliptic curve and finite Abelian group lattices.

Findings

Lattices are generated by their minimal vectors

Estimated total number of minimal vectors

Analyzed automorphism groups of these lattices

Abstract

We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of Gerhard Hiss on the factorization of functions with particular divisor support into lines and their inverses.