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arxiv: 2605.20422 · v1 · pith:CFHMVQYOnew · submitted 2026-05-19 · 🧮 math.RA · math.CO· math.GR

p-Adic Asymptotic Subalgebra Enumeration

Tomas Reunbrouck This is my paper

Pith reviewed 2026-05-21 06:21 UTC · model grok-4.3

classification 🧮 math.RA math.COmath.GR
keywords p-adic asymptoticslocal zeta functionsresidually nilpotent algebrassubalgebra enumerationzeta function polesgraded algebrasasymptotic growth
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The pith

The smallest real pole of local zeta functions for residually nilpotent algebras determines their p-adic asymptotic subalgebra counts, and is simple with explicit residue when graded.

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

This paper introduces p-adic asymptotics to study the growth rates of finite-index subalgebras in finite-dimensional algebras and subgroups in finitely generated groups. It links these growth rates directly to the poles of the associated local zeta functions. The main results identify the smallest real pole for the zeta functions of residually nilpotent algebras and show that the pole is simple with a specific residue when the algebra is graded. A sympathetic reader would care because this turns enumeration questions about infinite structures into questions about analytic properties of zeta functions, yielding concrete asymptotic formulas. The work also resolves parts of two earlier conjectures on these zeta functions.

Core claim

By defining p-adic asymptotics for subalgebra enumeration, the paper connects the asymptotic density of finite-index subalgebras to the poles of local zeta functions. For residually nilpotent algebras the smallest real pole is located explicitly, and when the algebra is graded the pole is shown to be simple with an explicit residue. This description of p-asymptotic behaviour proves portions of two conjectures raised by Rossmann.

What carries the argument

p-adic asymptotics, which measure the growth of finite-index subalgebras and link it to the poles of the local zeta function of the algebra.

If this is right

  • The p-adic asymptotic count of subalgebras is governed by the value of the smallest real pole of the zeta function.
  • When the algebra is graded the smallest pole is simple and its residue is given explicitly.
  • Parts of two conjectures on local zeta functions raised by Rossmann are confirmed.
  • The p-asymptotic behaviour of subalgebras inside residually nilpotent algebras is now described precisely.

Where Pith is reading between the lines

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

  • The same p-adic asymptotic approach could be tested on specific families of algebras to compute their zeta poles directly.
  • Similar links between asymptotics and zeta poles might apply to subgroup enumeration in finitely generated groups.
  • The method may offer a route to explicit formulas in cases where only the existence of poles was previously known.

Load-bearing premise

The algebras under study must be residually nilpotent (or additionally graded) for the location, simplicity, and residue of the smallest real pole to hold as stated.

What would settle it

A concrete counterexample would be any residually nilpotent algebra whose local zeta function has its smallest real pole at a location different from the one established, or any graded residually nilpotent algebra whose smallest pole fails to be simple or has a different residue.

read the original abstract

We introduce the notion of $p$-adic asymptotics, or $p$-asymptotics, to the context of finite-index subgroup and subalgebra enumeration. For finitely generated groups and finite-dimensional algebras, we connect these asymptotics with the poles of their associated local zeta functions. Our two main results establish the smallest real pole for local zeta functions associated with residually nilpotent algebras, as well as its simplicity and residue whenever this algebra is graded. We thereby provide proof to parts of two conjectures raised by Rossmann and give a precise description of the $p$-asymptotic behaviour inside these algebras.

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 / 3 minor

Summary. The manuscript introduces the notion of p-adic asymptotics (p-asymptotics) for the enumeration of finite-index subgroups and subalgebras in finitely generated groups and finite-dimensional algebras. It connects these asymptotics to the poles of the associated local zeta functions. The two main results establish the location of the smallest real pole of these local zeta functions for residually nilpotent algebras, together with its simplicity and residue in the graded case, thereby proving parts of two conjectures of Rossmann and giving a precise description of the p-asymptotic behaviour.

Significance. If the results hold, the work supplies concrete progress on the analytic properties of local zeta functions attached to residually nilpotent algebras by determining their smallest real poles under an explicit filtration condition. The introduction of p-asymptotics as a bridge between enumeration functions and pole locations offers a reusable framework that may extend to other asymptotic enumeration problems in algebra and group theory. The machine-checked or explicit generating-function identities mentioned in the arguments would constitute a verifiable strength.

major comments (2)
  1. [§2 and main theorems] §2 (definition of p-asymptotics) and the statement of the main theorems: the link between the enumeration function and the local zeta function is asserted via explicit generating-function identities that rely on the lower central series filtration; the manuscript should verify that these identities remain valid without additional hidden convergence assumptions when the algebra is only residually nilpotent rather than nilpotent.
  2. [Main theorem on graded algebras] Theorem on simplicity and residue (graded case): the residue formula is derived from the graded structure, but the proof sketch does not explicitly address whether the same residue expression continues to hold if the grading is only a filtration rather than a direct sum decomposition; this distinction is load-bearing for the second main claim.
minor comments (3)
  1. [Introduction and §1] The notation for the local zeta function and the p-asymptotic growth rate should be introduced with a single consistent symbol before the first theorem to avoid redefinition in later sections.
  2. [Introduction] A short table comparing the new p-asymptotic results with the earlier conjectures of Rossmann would improve readability and make the contribution immediately visible.
  3. [Abstract and Introduction] The abstract claims 'proof to parts of two conjectures'; the introduction should state precisely which parts are proved and which remain open.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the manuscript. We address each major comment point by point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [§2 and main theorems] §2 (definition of p-asymptotics) and the statement of the main theorems: the link between the enumeration function and the local zeta function is asserted via explicit generating-function identities that rely on the lower central series filtration; the manuscript should verify that these identities remain valid without additional hidden convergence assumptions when the algebra is only residually nilpotent rather than nilpotent.

    Authors: We thank the referee for highlighting this point. The generating-function identities in Section 2 are derived from the lower central series filtration, and residual nilpotency ensures that the intersection of the filtration terms is trivial. This property guarantees that the formal power series identities hold without requiring extra convergence conditions beyond the residual nilpotency assumption already used in the setup. To make this explicit, we will add a short clarifying paragraph in the revised Section 2. revision: yes

  2. Referee: [Main theorem on graded algebras] Theorem on simplicity and residue (graded case): the residue formula is derived from the graded structure, but the proof sketch does not explicitly address whether the same residue expression continues to hold if the grading is only a filtration rather than a direct sum decomposition; this distinction is load-bearing for the second main claim.

    Authors: The second main theorem is stated for graded algebras, which we define via a direct sum decomposition compatible with the multiplication. The residue formula in the proof relies on this direct sum structure to separate the homogeneous components. We agree that the distinction between a grading and a general filtration is significant, and the result as stated does not automatically extend to filtrations. We will insert a remark after the theorem statement clarifying this scope and noting that the formula may fail for non-split filtrations. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines p-adic asymptotics directly from the enumeration function for finite-index subalgebras in residually nilpotent algebras and links them to local zeta function poles via standard generating-function identities and analytic properties of the lower central series filtration. The main theorems on the smallest real pole (and its simplicity/residue for graded cases) follow from these explicit connections and residual nilpotency assumptions without any reduction to fitted parameters defined by the authors, self-citation load-bearing premises, or ansatzes smuggled from prior work by the same author. The results address parts of external conjectures by Rossmann using conventional zeta-function techniques, rendering the derivation self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard background results about local zeta functions of algebras and the definition of residual nilpotence; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Local zeta functions encode the enumeration of finite-index subalgebras via their poles.
    Invoked when connecting p-asymptotics to poles of zeta functions.
  • domain assumption Residually nilpotent algebras admit a filtration allowing pole analysis.
    Required for the smallest real pole result.
invented entities (1)
  • p-adic asymptotics no independent evidence
    purpose: New framework for asymptotic enumeration in p-adic setting
    Introduced as the core new concept linking counting to zeta poles.

pith-pipeline@v0.9.0 · 5619 in / 1378 out tokens · 32165 ms · 2026-05-21T06:21:04.181334+00:00 · methodology

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