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Counting wildly ramified quartic extensions with prescribed discriminant and Galois closure group

Sebastian Monnet

Abstract

Given a $2$-adic field $K$, we give formulae for the number of totally ramified quartic field extensions $L/K$ with a given discriminant valuation and Galois closure group. We use these formulae to prove a refinement of Serre's mass formula, which will have applications to the arithmetic statistics of number fields.

Counting wildly ramified quartic extensions with prescribed discriminant and Galois closure group

Abstract

Given a -adic field , we give formulae for the number of totally ramified quartic field extensions with a given discriminant valuation and Galois closure group. We use these formulae to prove a refinement of Serre's mass formula, which will have applications to the arithmetic statistics of number fields.
Paper Structure (16 sections, 60 theorems, 194 equations)

This paper contains 16 sections, 60 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Outline and key results
  3. Application: Refinements of Serre's mass formula
  4. Correctness of results
  5. Index of notation
  6. Acknowledgements
  7. The cases $G = S_4$ and $G = A_4$
  8. Congruence conditions for $P^{\mathrm{1-Aut}}_m$
  9. Computing the densities
  10. Distinguishing between $A_4$ and $S_4$
  11. The Case $G = V_4$
  12. The Case $G = C_4$
  13. Sketch of our approach
  14. Counting $C_4$-extendable extensions
  15. Counting $C_4$-extensions with a given intermediate field
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that $f(K/\mathbb{Q}_2)$ is even. Then $\Sigma_{m}^{S_4}$ is empty for all $m$. Moreover, $\Sigma_{m}^{A_4}$ is nonempty if and only if $m$ is an even integer with $4 \leq m \leq 6e_K + 2$. In that case, we have

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 114 more