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The Capitulation Problem in Certain Pure Cubic Fields

Siham Aouissi, Daniel C. Mayer

TL;DR

The study investigates capitulation in the normal closures $k=\mathbb{Q}(\sqrt[3]{n},\zeta)$ of pure cubic fields with $\mathrm{Cl}_3(k)\cong(\mathbb{Z}/3\mathbb{Z})^2$ under conductors of shape $f\in\{pq_1q_2,3pq,9pq\}$, challenging prior claims by Ismaili that only two capitulation types occur. By correcting the theoretical framework (via a proper Huppert-based argument) and analyzing unit-norm indices, the authors show four possible capitulation types $(1320),(0320),(1000),(0000)$, with two additional cases depending on a differential principal factorization (DPF) type $\alpha$ restricting to $(0000)$ or $(0320)$. Extensive GRH-based computations and coclass theory yield numerous counterexamples across Dedekind species II, IB, IA, confirming that the four CTs occur in practice and that higher coclasses are realized in the second $3$-class field tower, with the big tower group often metabelian $G\simeq\langle54,13\rangle$. The results correct the literature on capitulation in pure cubic fields, broaden the taxonomy of transfer patterns, and illuminate the structure of Hilbert $3$-class field towers in this setting.

Abstract

Let \(Γ=\mathbb{Q}(\sqrt[3]{n})\) be a pure cubic field with normal closure \(k=\mathbb{Q}(\sqrt[3]{n},ζ)\), where \(n>1\) denotes a cube free integer, and \(ζ\) is a primitive cube root of unity. Suppose \(k\) possesses an elementary bicyclic \(3\)-class group \(\mathrm{Cl}_3(k)\), and the conductor of \(k/\mathbb{Q}(ζ)\) has the shape \(f\in\lbrace pq_1q_2,3pq,9pq\rbrace\) where \(p\equiv 1\,(\mathrm{mod}\,9)\) and \(q,q_1,q_2\equiv 2,5\,(\mathrm{mod}\,9)\) are primes. It is disproved that there are only two possible capitulation types \(\varkappa(k)\), either type \(\mathrm{a}.1\), \((0000)\), or type \(\mathrm{a}.2\), \((1000)\). Evidence is provided, theoretically and experimentally, of two further types, \(\mathrm{b}.10\), \((0320)\), and \(\mathrm{d}.23\), \((1320)\).

The Capitulation Problem in Certain Pure Cubic Fields

TL;DR

The study investigates capitulation in the normal closures of pure cubic fields with under conductors of shape , challenging prior claims by Ismaili that only two capitulation types occur. By correcting the theoretical framework (via a proper Huppert-based argument) and analyzing unit-norm indices, the authors show four possible capitulation types , with two additional cases depending on a differential principal factorization (DPF) type restricting to or . Extensive GRH-based computations and coclass theory yield numerous counterexamples across Dedekind species II, IB, IA, confirming that the four CTs occur in practice and that higher coclasses are realized in the second -class field tower, with the big tower group often metabelian . The results correct the literature on capitulation in pure cubic fields, broaden the taxonomy of transfer patterns, and illuminate the structure of Hilbert -class field towers in this setting.

Abstract

Let \(Γ=\mathbb{Q}(\sqrt[3]{n})\) be a pure cubic field with normal closure \(k=\mathbb{Q}(\sqrt[3]{n},ζ)\), where denotes a cube free integer, and is a primitive cube root of unity. Suppose possesses an elementary bicyclic -class group \(\mathrm{Cl}_3(k)\), and the conductor of \(k/\mathbb{Q}(ζ)\) has the shape where \(p\equiv 1\,(\mathrm{mod}\,9)\) and \(q,q_1,q_2\equiv 2,5\,(\mathrm{mod}\,9)\) are primes. It is disproved that there are only two possible capitulation types \(\varkappa(k)\), either type , \((0000)\), or type , \((1000)\). Evidence is provided, theoretically and experimentally, of two further types, , \((0320)\), and , \((1320)\).
Paper Structure (14 sections, 7 theorems, 19 equations, 7 tables)

This paper contains 14 sections, 7 theorems, 19 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Clarifications on capitulation types
  3. Pure cubic fields of type III
  4. Capitulation kernels and abelian type invariants
  5. Little and big two-stage towers
  6. Theoretical proof
  7. Experimental proof
  8. Counterexamples of Dedekind species II
  9. Counterexamples of Dedekind species IB
  10. Counterexamples of Dedekind species IA
  11. Prototypes of counterexamples
  12. Experimental proof by coclass
  13. Data availability statement
  14. Acknowledgements

Key Result

Theorem 1

The structure of all unramified abelian $3$-extensions of $k$, viewed as Kummer extensions, is given by adjoining cube roots of the prime elements $\pi_1,\pi_2$ lying over the prime divisor $p\equiv 1\,(\mathrm{mod}\,9)$ of the conductor $f$: the relative genus field is bicyclic bicubic, and the four unramified cyclic cubic extensions of $k$ are given by Is1992 If the automorphism group $\mathrm{

Theorems & Definitions (22)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 2
  • Corollary 1
  • proof
  • Definition 4
  • Theorem 3
  • proof
  • ...and 12 more