Bases of associated Galois modules in general wildly ramified extensions and in elementary abelian extensions of degree $p^2$

Kirill S. Ladny; Konstantin I. Pimenov; Mikhail V. Bondarko

arxiv: 2511.20634 · v2 · pith:TJ6PIPJLnew · submitted 2025-11-25 · 🧮 math.AG · math.NT

Bases of associated Galois modules in general wildly ramified extensions and in elementary abelian extensions of degree p²

Mikhail V. Bondarko , Kirill S. Ladny , Konstantin I. Pimenov This is my paper

Pith reviewed 2026-05-22 12:28 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords wildly ramified extensionsGalois moduleselementary abelian extensionsramification jumpsvaluation filtrationgroup algebradiscrete valuation fields

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The pith

In wildly ramified extensions with distinct ramification jumps modulo p squared, elements of the group algebra form bases for associated Galois modules.

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

This paper studies collections of elements from the group algebra k[G] that can be used to build bases for various Galois modules and orders attached to wildly ramified extensions K/k of complete discrete valuation fields. In the special case G equals (Z/pZ) squared the authors compute the explicit action of the elements (σ1-1) to the i times (σ2-1) to the j on the valuation filtration of the extension. The central result states that whenever the ramification jumps are distinct modulo p squared these elements supply sufficiently good bases. A reader would care because such explicit bases make the integral structure of the extension more accessible for further calculations.

Core claim

For a wildly ramified extension K/k of complete discrete valuation fields the paper studies collections of elements of k[G] that fit well for constructing bases of associated Galois modules and orders. In the case G equals (Z/pZ) squared the action of the elements (σ1-1)^i (σ2-1)^j for 0 less than or equal to i,j less than or equal to p-1 on the valuation filtration is computed explicitly. If the ramification jumps of K/k are distinct modulo p squared then these elements yield good enough bases.

What carries the argument

The elements (σ1-1)^i (σ2-1)^j in the group algebra k[G] that act on the valuation filtration and generate bases when the ramification jumps are distinct modulo p squared.

If this is right

The same collections of elements from k[G] can be used to describe associated orders in the integral closure of the extension.
Explicit computation of the Galois-module structure becomes feasible whenever the ramification jumps satisfy the distinctness condition modulo p squared.
The method supplies a concrete way to find suitable bases in the elementary abelian case of degree p squared.
The approach extends the study from the elementary abelian case to more general wildly ramified extensions by identifying workable collections of group-algebra elements.

Where Pith is reading between the lines

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

The same technique of computing actions on the valuation filtration may apply to abelian extensions of higher p-power degree once analogous jump conditions are identified.
These bases could be used to calculate local Artin conductors or conductors of characters in an explicit manner.
The results connect to the broader theory of orders in group algebras and may help classify their structures in ramified settings.
Concrete numerical checks in p-adic fields or function fields over finite fields of characteristic p would test the condition on ramification jumps.

Load-bearing premise

The assumption that ramification jumps distinct modulo p squared suffice to make the chosen elements produce good bases, together with the explicit action computation for the given generators σ1 and σ2.

What would settle it

An explicit example of a wildly ramified extension K/k with Galois group (Z/pZ) squared where the ramification jumps are distinct modulo p squared yet the elements (σ1-1)^i (σ2-1)^j fail to form a basis for one of the associated Galois modules.

read the original abstract

For a wildly ramified extension $K/k$ of complete discrete valuation fields we study collections of elements of $k[G]$ (where $G=Gal(K/k)$) that fit well for constructing bases of various associated Galois modules and orders. In the case $G=(Z/pZ)^2$ (where $p$ is the characteristic of residue fields) we are able to compute the action of the elements $(\sigma_1-1)^i(\sigma_2-1)^j,\ 0\le i,j\le p-1,$ on the valuation filtration; here $\sigma_1,\sigma_2$ are generators of $G$. If the ramification jumps of $K/k$ are distinct modulo $p^2$ then these elements do yield "good enough" bases in question.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit formulas for how (σ1-1)^i(σ2-1)^j act on the valuation filtration in elementary abelian p²-extensions and claims these yield good bases when jumps are distinct mod p².

read the letter

The main takeaway is the concrete computation of the action of those group-ring elements on the filtration for G=(Z/pZ)², plus the conditional claim that distinct ramification jumps mod p² make them good enough for bases of the associated modules and orders. That explicit description for the two generators is the actual new piece here; prior general theory on ramification does not usually hand you the matrix of actions in this form. The setup for complete discrete valuation fields is handled cleanly and the focus stays on the elementary abelian case, which keeps the work targeted. The soft spot is the jump from the action formulas to the basis conclusion. The paper uses the distinct-jumps-mod-p² condition to assert that the elements produce the desired bases, but it is not obvious from the abstract alone how the filtration actions guarantee spanning or independence without extra relations from the ramification filtration or the precise placement of the jumps. If the full text derives this step directly from the computed actions, the argument holds; otherwise that sufficiency needs more detail. This is for specialists in local number theory and ramification theory who need explicit bases for computations in wildly ramified p-extensions. A reader working on Galois modules in positive characteristic or p-adic fields could use the formulas. It deserves a serious referee to check the derivations and the basis step.

Referee Report

1 major / 0 minor

Summary. The manuscript examines collections of elements in the group algebra k[G] for G = Gal(K/k) in wildly ramified extensions K/k of complete discrete valuation fields, with the goal of identifying those that serve as suitable bases for associated Galois modules and orders. In the elementary abelian case G ≅ (ℤ/pℤ)², the authors compute the action of the elements (σ₁ − 1)^i (σ₂ − 1)^j for 0 ≤ i, j ≤ p − 1 on the valuation filtration, where σ₁ and σ₂ generate G. They conclude that if the ramification jumps of K/k are distinct modulo p², then these elements yield 'good enough' bases for the modules under consideration.

Significance. If the central computations and conditional basis statement hold, the work supplies explicit, computable bases for Galois modules in p²-extensions with controlled ramification jumps. This could support further study of integral structures and ramification in arithmetic geometry. The explicit action formulas on the filtration constitute a concrete strength that permits direct checks in examples.

major comments (1)

[Abstract] Abstract / central claim: The transition from the computed action of (σ₁ − 1)^i (σ₂ − 1)^j on the valuation filtration to the conclusion that these elements form 'good enough' bases (i.e., generate the required modules with the stated independence) when ramification jumps are distinct modulo p² is asserted rather than derived. The manuscript does not explicitly exhibit the spanning or freeness argument from the action matrix, leaving open the possibility that further relations in the ramification filtration or the precise embedding of the jumps could introduce dependencies. This step is load-bearing for the main result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comment on the central claim is well taken, and we address it directly below.

read point-by-point responses

Referee: [Abstract] Abstract / central claim: The transition from the computed action of (σ₁ − 1)^i (σ₂ − 1)^j on the valuation filtration to the conclusion that these elements form 'good enough' bases (i.e., generate the required modules with the stated independence) when ramification jumps are distinct modulo p² is asserted rather than derived. The manuscript does not explicitly exhibit the spanning or freeness argument from the action matrix, leaving open the possibility that further relations in the ramification filtration or the precise embedding of the jumps could introduce dependencies. This step is load-bearing for the main result.

Authors: We agree that the passage from the explicit action formulas to the basis statement requires a more transparent derivation. The computations in the paper determine the leading terms of the action on successive quotients of the valuation filtration. When the ramification jumps are distinct modulo p², these leading coefficients produce a matrix (with respect to the natural filtration basis) that is upper-triangular with diagonal entries that are nonzero in the residue field, hence invertible over the ring of integers. This invertibility directly implies both spanning and freeness, ruling out additional relations. In the revised manuscript we will add a short subsection immediately after the action computations that constructs this matrix explicitly and verifies its invertibility under the stated hypothesis. The original computations remain unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on explicit computations

full rationale

The paper computes the action of the elements (σ₁−1)^i(σ₂−1)^j on the valuation filtration for G=(Z/pZ)^2 and states that distinct ramification jumps modulo p² suffice for these to yield good bases. This chain is presented as direct verification from the action formulas and the jump condition, without reduction to fitted inputs, self-definitional loops, or load-bearing self-citations. The central claims remain independent of the paper's own outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions of local number theory concerning complete discrete valuation fields and the definition of ramification jumps; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption K/k is a wildly ramified extension of complete discrete valuation fields with Galois group G.
This is the setting stated in the abstract for studying the Galois modules and the action on the valuation filtration.

pith-pipeline@v0.9.0 · 5687 in / 1341 out tokens · 63100 ms · 2026-05-22T12:28:05.597861+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Math., vol

Bondarko M.V., Local Leopoldt's problem for rings of integers in abelian p-extensions of complete discrete valuation fields// Doc. Math., vol. 5, 657--693, 2000

work page 2000
[2]

Bondarko M.V., Local Leopoldt's problem for ideals in p-extensions of complete discrete valuation fields// Algebraic Number Theory and Algebraic Geometry: Papers Dedicated to A. N. Parshin on the Occasion of his Sixtieth Birthday, 27--57, Contemporary Mathematics, Providence, 2002

work page 2002
[3]

of Number Theory, vol

Bondarko M.V., Links between associated additive Galois modules and computation of H^1 for local formal group modules// J. of Number Theory, vol. 101, 74--104, 2003

work page 2003
[4]

9(1), 201--219, 1997

Byott N., Galois structure of ideals in wildly ramified abelian p -extensions of a p -adic field, and some applications // Journal de Theorie des nombres de Bordeaux, v. 9(1), 201--219, 1997

work page 1997
[5]

Byott N., Associated orders of certain extensions arising from Lubin-Tate formal groups // Journal de Theorie des nombres de Bordeaux. 1997. v. 9, 449--462, 1997

work page 1997
[6]

121, American Mathematical Soc., 2002

Fesenko I.B., Vostokov S.V., Local Fields and their extensions, second edition, Translations of Mathematical Monographs vol. 121, American Mathematical Soc., 2002

work page 2002
[7]

Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlk\

Leopoldt H.-W., \"Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlk\"orpers. (German), J. Reine Angew. Math. 201 (1959), 119--149

work page 1959
[8]

Graduate texts in Mathematics, vol

Serre J.-P., Local fields. Graduate texts in Mathematics, vol. 67, Springer Science & Business Media, 1979

work page 1979
[9]

Bondarko M.V., Leopoldt's problem for totally ramified extensions of complete discrete valuation fields// Algebra and Analysis, 18(5), 2006, 99--129, transl. in: St. Petersburg Math. J. 18 (2007), 757--778

work page 2006

[1] [1]

Math., vol

Bondarko M.V., Local Leopoldt's problem for rings of integers in abelian p-extensions of complete discrete valuation fields// Doc. Math., vol. 5, 657--693, 2000

work page 2000

[2] [2]

Bondarko M.V., Local Leopoldt's problem for ideals in p-extensions of complete discrete valuation fields// Algebraic Number Theory and Algebraic Geometry: Papers Dedicated to A. N. Parshin on the Occasion of his Sixtieth Birthday, 27--57, Contemporary Mathematics, Providence, 2002

work page 2002

[3] [3]

of Number Theory, vol

Bondarko M.V., Links between associated additive Galois modules and computation of H^1 for local formal group modules// J. of Number Theory, vol. 101, 74--104, 2003

work page 2003

[4] [4]

9(1), 201--219, 1997

Byott N., Galois structure of ideals in wildly ramified abelian p -extensions of a p -adic field, and some applications // Journal de Theorie des nombres de Bordeaux, v. 9(1), 201--219, 1997

work page 1997

[5] [5]

Byott N., Associated orders of certain extensions arising from Lubin-Tate formal groups // Journal de Theorie des nombres de Bordeaux. 1997. v. 9, 449--462, 1997

work page 1997

[6] [6]

121, American Mathematical Soc., 2002

Fesenko I.B., Vostokov S.V., Local Fields and their extensions, second edition, Translations of Mathematical Monographs vol. 121, American Mathematical Soc., 2002

work page 2002

[7] [7]

Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlk\

Leopoldt H.-W., \"Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlk\"orpers. (German), J. Reine Angew. Math. 201 (1959), 119--149

work page 1959

[8] [8]

Graduate texts in Mathematics, vol

Serre J.-P., Local fields. Graduate texts in Mathematics, vol. 67, Springer Science & Business Media, 1979

work page 1979

[9] [9]

Bondarko M.V., Leopoldt's problem for totally ramified extensions of complete discrete valuation fields// Algebra and Analysis, 18(5), 2006, 99--129, transl. in: St. Petersburg Math. J. 18 (2007), 757--778

work page 2006