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

Lilian Menn, Elif Sacikara

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 Böttcher et al., 2016.

Successive Minima, Determinant and Automorphism Groups of Hyperelliptic Function Field 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 Böttcher et al., 2016.

Paper Structure

This paper contains 12 sections, 28 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperelliptic Function Field Lattices
  4. Successive Minima
  5. Determinant
  6. Automorphisms
  7. General Algebraic Function Field Lattices
  8. Rational Function Field Lattices
  9. Lattices from Finite Abelian Groups
  10. Elliptic Function Field Lattices
  11. Hermitian Function Field Lattices
  12. Hyperelliptic Function Field Lattices

Key Result

Proposition 2.2

atecs2017lattices Let $\mathcal{P}$ and $\mathcal{O}_{\mathcal{P}}^*$ be as in Definition defn:FuncFieldLattice, then the following holds.

Theorems & Definitions (51)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Example 2.4: Rational Function Field Lattices
  • Example 2.5: Elliptic Function Field Lattices
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • ...and 41 more