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Initial layer of the anti-cyclotomic $\mathbb{Z}_3$-extension of $\mathbb{Q}(\sqrt{-m})$ and capitulation phenomenon

Georges Gras

TL;DR

This work studies capitulation of the $p$-class group in the anti-cyclotomic ${Z}_p$-extension ${k^{ac}}$ of an imaginary quadratic field ${k}$, introducing a novel ${Log}_p$-based method to determine the first layer ${k_1^{ac}}$ and to analyze partial capitulation. It develops a detailed arithmetic framework for the ${Lg}$-extension, including unit groups, the Artin group, and minus/plus decompositions, and then applies Kummer theory through characters and radicals to describe ramification and reflection phenomena. The paper provides explicit results (Theorems A–G) linking capitulation to invariants like ${H_k}$, ${T_k}$, and the normalized $3$-adic regulator ${R_{k^*}}$, especially in the Normal Split and Special Split cases for $p=3$, and supplies four PARI/GP programs to compute the defining polynomials of ${k_1^{ac}}$ and related invariants. It also offers a critical examination of prior assertions about Lg in cyclic cases, clarifying when linear disjunction occurs and when capitulation can be expected, and connects the computational findings to Iwasawa theory themes, suggesting Greenberg-like behavior for non-cyclotomic ${Z}_p$-extensions. Overall, the work demonstrates that non-cyclotomic ${Z}_p$-extensions of imaginary quadratic fields exhibit capitulation patterns akin to cyclotomic situations, with computable first layers and accessible relationships to Iwasawa invariants via the Log_p framework.

Abstract

Let $k=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic field. We consider the properties of capitulation of the $p$-class group of $k$ in the anti-cyclotomic $\mathbb{Z}_p$-extension $k^{\rm ac}$ of $k$; for this, using a new approach based on the Log$_p$-function (Theorems 2.3, 3.4), we determine the first layer $k_1^{\rm ac}$ of $k^{\rm ac}$ over $k$, and we show that some partial capitulation may exist in $k_1^{\rm ac}$, even when $k^{\rm ac}/k$ is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the $\mathbb{Z}_p$-extensions of $k$, distinct from the cyclotomic one. For $p=3$, we characterize a sub-family of fields $k$ (Normal Split cases) for which $k^{\rm ac}$ is not linearly disjoint from the Hilbert class field (Theorem 5.1). No assumptions are made on the splitting of 3 in $k$ and in $k^*=\mathbb{Q}(\sqrt{3m})$, nor on the structures of their 3-class groups. Four PARI/GP programs (7.1, 7.2, 7.3, 7.4 depending on the classification of Definition 2.10) are given, computing a defining cubic polynomial of $k_1^{\rm ac}$, and the main invariants attached to the fields $k$, $k^*$, $k_1^{\rm ac}$; some relations with Iwasawa's invariants are discussed (Theorem 9.6).

Initial layer of the anti-cyclotomic $\mathbb{Z}_3$-extension of $\mathbb{Q}(\sqrt{-m})$ and capitulation phenomenon

TL;DR

This work studies capitulation of the -class group in the anti-cyclotomic -extension of an imaginary quadratic field , introducing a novel -based method to determine the first layer and to analyze partial capitulation. It develops a detailed arithmetic framework for the -extension, including unit groups, the Artin group, and minus/plus decompositions, and then applies Kummer theory through characters and radicals to describe ramification and reflection phenomena. The paper provides explicit results (Theorems A–G) linking capitulation to invariants like , , and the normalized -adic regulator , especially in the Normal Split and Special Split cases for , and supplies four PARI/GP programs to compute the defining polynomials of and related invariants. It also offers a critical examination of prior assertions about Lg in cyclic cases, clarifying when linear disjunction occurs and when capitulation can be expected, and connects the computational findings to Iwasawa theory themes, suggesting Greenberg-like behavior for non-cyclotomic -extensions. Overall, the work demonstrates that non-cyclotomic -extensions of imaginary quadratic fields exhibit capitulation patterns akin to cyclotomic situations, with computable first layers and accessible relationships to Iwasawa invariants via the Log_p framework.

Abstract

Let be an imaginary quadratic field. We consider the properties of capitulation of the -class group of in the anti-cyclotomic -extension of ; for this, using a new approach based on the Log-function (Theorems 2.3, 3.4), we determine the first layer of over , and we show that some partial capitulation may exist in , even when is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the -extensions of , distinct from the cyclotomic one. For , we characterize a sub-family of fields (Normal Split cases) for which is not linearly disjoint from the Hilbert class field (Theorem 5.1). No assumptions are made on the splitting of 3 in and in , nor on the structures of their 3-class groups. Four PARI/GP programs (7.1, 7.2, 7.3, 7.4 depending on the classification of Definition 2.10) are given, computing a defining cubic polynomial of , and the main invariants attached to the fields , , ; some relations with Iwasawa's invariants are discussed (Theorem 9.6).

Paper Structure

This paper contains 53 sections, 25 theorems, 63 equations, 2 algorithms.

Table of Contents

  1. Initial layer of the Lg-extension Lg of Lg
  2. Introduction
  3. Main statements
  4. The arithmetic of Lg -- Algorithms
  5. Unit groups of the layers of Lg
  6. Artin group of the first layer of a Lg-extension
  7. Minus and plus components attached to Lg
  8. Lg-ranks of ray class fields
  9. Technical conditions for Lg
  10. Description of the method Lg
  11. Kummer radicals and characters
  12. Characters of Lg
  13. Units, pseudo-units and Lg-units -- Radicals
  14. Classification of the Kummer radicals
  15. Abelian Lg-ramification over Lg
  16. ...and 38 more sections

Key Result

Lemma 1.1

Let ${\mathcal{H}}$ be a finite abelian $p$-group, and let ${\mathcal{H}}_0 = \langle h_0 \rangle$ be a cyclic subgroup of ${\mathcal{H}}$. Then $\overline {\mathcal{H}}_0 := {\mathcal{H}}_0 \cdot {\mathcal{H}}^p/{\mathcal{H}}^p$ is a non-trivial direct factor of $\overline {\mathcal{H}} := {\mathc

Theorems & Definitions (56)

  • Lemma 1.1
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.3
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • proof
  • Proposition 2.9
  • ...and 46 more