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arxiv: 2403.16603 · v6 · submitted 2024-03-25 · 🧮 math.NT

On the mathbb{Z}_p-extensions of a totally p-adic imaginary quadratic field -- With an appendix by Jean-Franc{c}ois Jaulent

Georges Gras (LMB) This is my paper

Pith reviewed 2026-05-24 03:38 UTC · model grok-4.3

classification 🧮 math.NT
keywords Z_p-extensionsimaginary quadratic fieldslogarithmic class groupFermat quotientp-class groupanti-cyclotomic extensioncapitulationIwasawa invariants
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The pith

The p-valuation of a Fermat quotient of the fundamental p-unit governs the Z_p-extensions and logarithmic class groups of totally p-adic imaginary quadratic fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that in an imaginary quadratic field k where an odd prime p splits completely, the p-adic valuation of the Fermat quotient associated to its fundamental p-unit determines the order of the logarithmic class group of k. This same valuation controls the structure of the non-cyclotomic Z_p-extensions of k, including the indices in the filtration of p-class groups along the tower, and it generalizes the Gold-Sands criterion without needing assumptions of total ramification or trivial class group. The results provide explicit relations for the first two layers of the filtration and for capitulation in the anti-cyclotomic extension, all derived from this single invariant. These findings offer new ways to compute class group behavior in infinite towers using only base field data.

Core claim

The central discovery is that the p-valuation δ_p(k) of a Fermat quotient of the fundamental p-unit x of k determines the order of the logarithmic class group H_k and the ratios #(H_{K_n}^2 / H_{K_n}^1) = #~H_k for large n in the Z_p-extension K/k, while also generalizing criteria for the p-class groups in these extensions and in the anti-cyclotomic subextension, without assuming total ramification or trivial p-class group.

What carries the argument

the p-valuation δ_p(k) of the Fermat quotient of the fundamental p-unit x of k, which serves as the governing arithmetic invariant linking units to class group filtrations in the extensions

If this is right

  • The order of the logarithmic class group is given explicitly in terms of δ_p(k).
  • The filtration quotients in the tower stabilize to a value determined by the base invariant for sufficiently large layers.
  • The Gold-Sands criterion is generalized to cases without total ramification.
  • Capitulation of suitable classes occurs in the first layer of the anti-cyclotomic Z_p-extension for p=3.
  • Large λ-invariants are realized in certain Z_p-extensions by choosing appropriate base fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If δ_p(k) can be computed algorithmically for many k, then the class group behavior in the entire tower becomes predictable from finite data.
  • The appendix suggests similar control may hold for abelian extensions of higher degree.
  • Conjecture 7.10 on further capitulation could be tested by extending the computations in Section 9 to more fields.

Load-bearing premise

The logarithmic class group H_k and the filtrations H_{K_n}^i are assumed to satisfy the stated relations with the Fermat quotient valuation from the beginning.

What would settle it

Finding a specific imaginary quadratic field k and prime p splitting in k where the computed order of the logarithmic class group does not match the value predicted from the Fermat quotient valuation of its fundamental p-unit.

read the original abstract

Let $k = \mathbb{Q}(\sqrt {-m})$ and $p \geq 3$ split in $k$. We prove new properties of the $\mathbb{Z}_p$-extensions $K/k$, distinct from the cyclotomic one; we do not assume $K/k$ totally ramified, nor the triviality of the $p$-class group of $k$. These properties are governed by the $p$-valuation $\delta_p(k)$ of a Fermat quotient of the fundamental $p$-unit $x$ of $k$, which also yields the order of the logarithmic class group $\# \mathcal{H}_k$ (Thm. 4.2 extended in App.A to the case of imaginary abelian fields of prime-to-$p$ degree), and allows to generalize the Gold-Sands criterion (Sec. 7). These results are related to the first two elements, $\mathcal{H}_{K_n}^1$ and $\mathcal{H}_{K_n}^2$, of the filtrations of the $p$-class groups in $K = \cup_n K_n$, without any argument of Iwasawa's theory, and provide new perspectives since $\# ( \mathcal{H}_{K_n}^2/ \mathcal{H}_{K_n}^1) = \# \widetilde {\mathcal{H}}_k$ for $n$ large enough (Thm 7.1). We give a short proof generalizing a result of Kundu-Washington (Thm. 7.8) on the $p$-class groups in the anti-cyclotomic $\mathbb{Z}_p$-extension $k^{\rm ac}$. We compute, Sec. 9, for $p = 3$, the first layer $k_1^{\rm ac}$ of $k^{\rm ac}$, using the Log$_p$-function, and show (Thms. 9.2,9.4) that capitulation of suitable ``classes'' is possible in $k^{\rm ac}$, suggesting Conjecture 7.10. Finally, we generalize (Thms.10.1,10.7) a result of Ozaki giving large $\lambda$'s invariants. Calculations and programs are gathered App. C.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to establish new properties of the non-cyclotomic ℤ_p-extensions K/k of an imaginary quadratic field k=ℚ(√−m) with p≥3 splitting in k. These properties are governed by the p-valuation δ_p(k) of the Fermat quotient of the fundamental p-unit x of k. Specifically, δ_p(k) determines the order of the logarithmic class group #ℋ_k (Theorem 4.2, extended in the appendix to imaginary abelian fields of prime-to-p degree), generalizes the Gold-Sands criterion, and controls the relation #(ℋ_{K_n}^2 / ℋ_{K_n}^1) = #~ℋ_k for sufficiently large n (Theorem 7.1). The results are obtained without assuming total ramification of K/k or triviality of the p-class group of k, and without employing Iwasawa theory. Additional results include a generalization of a theorem of Kundu-Washington on the anti-cyclotomic extension (Theorem 7.8), explicit computations for p=3 using the Log_p function (Theorems 9.2, 9.4), and generalizations of Ozaki's results on large λ-invariants (Theorems 10.1, 10.7).

Significance. If the claimed linkages between δ_p(k) and the logarithmic class group filtrations hold, the paper would provide a novel approach to studying p-class groups in ℤ_p-extensions that bypasses standard Iwasawa-theoretic machinery, offering explicit criteria based on Fermat quotients and computational verifiability through the programs in Appendix C. This could be significant for understanding capitulation phenomena and class number growth in such extensions. The inclusion of an appendix by Jaulent extending one of the main theorems adds credibility to the abelian case generalization.

major comments (2)
  1. [§4, Theorem 4.2] §4, Theorem 4.2: The theorem asserts that δ_p(k) yields #ℋ_k, but the manuscript invokes the existence, basic properties, and precise linkage of the logarithmic class group ℋ_k to the Fermat quotient valuation δ_p(k) without deriving this connection internally; the proof of the theorem is not visible in the text.
  2. [§7, Theorem 7.1] §7, Theorem 7.1: The claim that #(ℋ_{K_n}^2 / ℋ_{K_n}^1) = #~ℋ_k for n large enough is presented as following from δ_p(k), yet the relation between the filtration quotients and the logarithmic class group is presupposed rather than established within the paper, particularly the avoidance of Iwasawa theory arguments.
minor comments (2)
  1. Notation for the logarithmic class group alternates between ℋ_k, H_k, and ~ℋ_k; consistent use throughout would improve clarity.
  2. [Appendix C] Appendix C mentions programs and calculations but does not specify the programming language or software environment used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on our manuscript. The points raised concern the visibility and internal derivation of key linkages in Theorems 4.2 and 7.1. We address each below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [§4, Theorem 4.2] §4, Theorem 4.2: The theorem asserts that δ_p(k) yields #ℋ_k, but the manuscript invokes the existence, basic properties, and precise linkage of the logarithmic class group ℋ_k to the Fermat quotient valuation δ_p(k) without deriving this connection internally; the proof of the theorem is not visible in the text.

    Authors: We agree that the step-by-step derivation linking δ_p(k) to #ℋ_k should be made fully explicit and self-contained within Section 4. The current text relies on the definition of the logarithmic class group and the Fermat quotient but does not spell out every intermediate equality. In the revised version we will expand the proof of Theorem 4.2 with a detailed chain of equalities showing how the p-valuation of the Fermat quotient determines the order, without external appeals for the core linkage. revision: yes

  2. Referee: [§7, Theorem 7.1] §7, Theorem 7.1: The claim that #(ℋ_{K_n}^2 / ℋ_{K_n}^1) = #~ℋ_k for n large enough is presented as following from δ_p(k), yet the relation between the filtration quotients and the logarithmic class group is presupposed rather than established within the paper, particularly the avoidance of Iwasawa theory arguments.

    Authors: The manuscript intends to derive the equality #(ℋ_{K_n}^2 / ℋ_{K_n}^1) = #~ℋ_k directly from the value of δ_p(k) and the explicit definition of the filtration on the p-class groups of the layers K_n, without Iwasawa theory. We acknowledge that the passage from the logarithmic class group to the filtration quotients is not written out with sufficient intermediate steps. The revised proof of Theorem 7.1 will insert these steps, showing how the relation follows from the earlier results on δ_p(k) and the filtration definitions while preserving the non-Iwasawa approach. revision: yes

Circularity Check

0 steps flagged

No circularity: central claims derive from δ_p(k) valuation applied to standard class-field objects

full rationale

The paper states Thm. 4.2 and Thm. 7.1 as consequences of the p-valuation δ_p(k) of the Fermat quotient of the fundamental p-unit, together with the existence of the logarithmic class group H_k and the filtrations H_{K_n}^i. These objects and their linkage are treated as given inputs from class-field theory rather than being redefined or fitted inside the paper; the theorems then compute orders and equalities from that valuation. No step reduces a prediction to a fitted parameter by construction, no self-citation chain is load-bearing for the governance claim, and no ansatz is smuggled via prior work of the same author. The derivation chain therefore remains self-contained against external benchmarks once the standard objects are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of the logarithmic class group H_k, the definition of the Fermat quotient valuation δ_p(k), and the standard properties of Z_p-extensions and p-class group filtrations in algebraic number theory; no new entities are postulated and no numerical parameters are fitted.

axioms (2)
  • domain assumption Existence and basic functoriality of the logarithmic class group H_k for imaginary quadratic fields
    Invoked in Thm. 4.2 and extended in App. A; treated as background from prior literature.
  • domain assumption Standard properties of Z_p-extensions and the filtration of p-class groups H_{K_n}^i
    Used throughout Sec. 7 and Thm. 7.1 without re-derivation.

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Works this paper leans on

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