Successive Minima, Determinant and Automorphism Groups of Hyperelliptic Function Field Lattices

TL;DR

This paper investigates the geometric properties of hyperelliptic function field lattices, focusing on successive minima, determinants, and automorphism groups, thereby extending understanding of their structure and symmetries.

Contribution

It provides new detailed analysis of successive minima and determinants of hyperelliptic function field lattices and links automorphism groups of algebraic function fields to these lattices.

Findings

Detailed characterization of successive minima.

Explicit formulas or bounds for determinants.

Established connection between automorphism groups and lattices.

Abstract

In this paper, we contribute to previously known results on lattices constructed by algebraic function fields, or function field lattices in short. First, motivated by the non-well-roundedness property of certain hyperelliptic function field lattices (Ates and Stichtenoth, 2016), we explore the successive minima of these lattices in detail. We also study the determinant of hyperelliptic function field lattices. Finally, we show a connection between the automorphism groups of algebraic function fields and function field lattices, based on ideas from B\"ottcher et al., 2016.