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Governing fields for hyperelliptic function fields

Joppe Stokvis

Abstract

We study the 8-rank of class groups of hyperelliptic function fields and show that such 8-ranks are governed by splitting conditions in so-called governing fields. A similar result was proven for quadratic number fields by Stevenhagen, who used a theory of Rédei symbols and Rédei reciprocity to do so. We introduce a version of the Rédei reciprocity law for function fields and use this to show existence of governing fields.

Governing fields for hyperelliptic function fields

Abstract

We study the 8-rank of class groups of hyperelliptic function fields and show that such 8-ranks are governed by splitting conditions in so-called governing fields. A similar result was proven for quadratic number fields by Stevenhagen, who used a theory of Rédei symbols and Rédei reciprocity to do so. We introduce a version of the Rédei reciprocity law for function fields and use this to show existence of governing fields.

Paper Structure

This paper contains 18 sections, 16 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline
  4. Notation
  5. Preliminaries
  6. Genus theory and Rédei theory
  7. The 2-rank
  8. The 4-rank
  9. The 8-rank
  10. Minimally ramified extensions and Rédei reciprocity
  11. Minimally ramified extensions
  12. The Rédei symbol
  13. Rédei reciprocity
  14. Governing fields
  15. Further research
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $D \in \mathbb{F}_q[x]$ be a squarefree polynomial. Then $\Omega_2(D)$ and $\Omega_4(D)$ exist. If not all irreducible factors of $D$ have even degree, then $\Omega_8(D)$ exists as well.

Figures (1)

  • Figure 1: : The field extensions used in \ref{['lem: local reciprocity']}.

Theorems & Definitions (43)

  • Definition 1.1: Governing fields
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Rosen
  • Theorem 3.1: Artin, Zhang, Semirat
  • proof : Proof (sketch)
  • Remark 3.2
  • Theorem 3.3: Wittmann
  • Theorem 4.1
  • Lemma 4.2
  • ...and 33 more