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arxiv: 1909.01764 · v5 · submitted 2019-09-04 · 🧮 math.NT

On the Iwasawa invariants of Kato's zeta elements for modular forms

Chan-Ho Kim , Jaehoon Lee , Gautier Ponsinet This is my paper

Pith reviewed 2026-05-24 15:22 UTC · model grok-4.3

classification 🧮 math.NT
keywords Iwasawa invariantsKato's zeta elementsmodular formsmain conjecturecongruencesIwasawa theoryp-adic L-functions
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The pith

Kato's main conjecture for higher weight modular forms propagates under congruences if zeta elements are non-vanishing mod p.

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

This paper studies the Iwasawa invariants of modules appearing in Kato's main conjecture for modular forms in the absence of p-adic L-functions. It analyzes how these invariants behave when modular forms are congruent to each other. The analysis generalizes several earlier results on similar invariants. As a result, the main conjecture is shown to propagate from base cases to higher weight forms at good primes, provided the zeta elements do not vanish modulo p. The findings also apply to signed and sharp-flat versions of Iwasawa theory for these forms.

Core claim

We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without p-adic L-functions under congruences. As a consequence, we establish the propagation of Kato's main conjecture for modular forms of higher weight at arbitrary good prime under the assumption on the mod p non-vanishing of Kato's zeta elements. The application to the ± and ♯/♭-Iwasawa theory for modular forms is also discussed.

What carries the argument

The Iwasawa modules attached to Kato's zeta elements for modular forms, whose invariants are tracked under congruences between forms.

If this is right

  • The main conjecture propagates to higher weight modular forms at arbitrary good primes.
  • The result applies directly to the plus-minus Iwasawa theory for modular forms.
  • The result applies directly to the sharp and flat Iwasawa theory for modular forms.
  • Prior results on Iwasawa invariants under congruences extend to this setting without p-adic L-functions.

Where Pith is reading between the lines

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

  • Verification of the main conjecture could reduce to weight-two cases where non-vanishing is easier to check computationally.
  • The same propagation technique might apply when p-adic L-functions are present or in other zeta-element constructions.
  • Explicit computation of invariants for known congruent families of forms could confirm or refute the non-vanishing hypothesis in practice.

Load-bearing premise

Kato's zeta elements are non-vanishing modulo p for the modular forms in question.

What would settle it

A pair of congruent modular forms of different weights where the zeta element is non-vanishing mod p but the Iwasawa invariants fail to match as required by the propagation relation.

read the original abstract

We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without $p$-adic $L$-functions under congruences. It generalizes the work of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim, Greenberg-Iovita-Pollack, and one of us simultaneously. As a consequence, we establish the propagation of Kato's main conjecture for modular forms of higher weight at arbitrary good prime under the assumption on the mod $p$ non-vanishing of Kato's zeta elements. The application to the $\pm$ and $\sharp/\flat$-Iwasawa theory for modular forms is also discussed.

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 examines the Iwasawa invariants of the modules appearing in Kato's main conjecture (without p-adic L-functions) under congruences between modular forms. It generalizes prior results of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim, and Greenberg-Iovita-Pollack to higher-weight forms and arbitrary good primes. As a consequence, it establishes propagation of Kato's main conjecture for such forms under the standing assumption of mod p non-vanishing of Kato's zeta elements, and discusses applications to the ± and sharp/flat Iwasawa theories.

Significance. If the derivations hold, the work supplies a systematic congruence-propagation mechanism that extends the reach of Kato's main conjecture to higher weights and all good primes, conditional on the explicit non-vanishing hypothesis. This mirrors and broadens the utility of earlier congruence techniques in the literature while remaining within the scope of the stated assumption.

major comments (2)
  1. [§4, Theorem 4.1] §4, Theorem 4.1: the reduction from the higher-weight case to the weight-2 case via the congruence map on zeta elements appears to rely on the compatibility of the Iwasawa modules under the Hecke action; the precise identification of the characteristic ideals after base change is not spelled out in sufficient detail to verify that no extra p-torsion is introduced.
  2. [§5.2, Proposition 5.3] §5.2, Proposition 5.3: the claim that the μ-invariant vanishes for the propagated module when it vanishes for the base form uses the non-vanishing hypothesis only at the level of the mod p reduction; it is unclear whether the argument controls the higher p-power torsion that could appear in the Iwasawa module for weight >2.
minor comments (2)
  1. [§6] Notation for the signed and sharp/flat Selmer groups in §6 is introduced without an explicit comparison table to the earlier literature (e.g., Kobayashi, Pollack); a short dictionary would improve readability.
  2. [Theorem 1.1] The statement of the main propagation result (Theorem 1.1) lists the non-vanishing hypothesis but does not indicate whether it is known to hold for any explicit infinite family of higher-weight forms at good primes; adding a remark on known cases would clarify the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised concern the level of detail in the proofs of Theorem 4.1 and Proposition 5.3. We address each below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [§4, Theorem 4.1] §4, Theorem 4.1: the reduction from the higher-weight case to the weight-2 case via the congruence map on zeta elements appears to rely on the compatibility of the Iwasawa modules under the Hecke action; the precise identification of the characteristic ideals after base change is not spelled out in sufficient detail to verify that no extra p-torsion is introduced.

    Authors: The compatibility of the Iwasawa modules under the Hecke action is established in Section 3 (specifically, the exact sequence in Proposition 3.4 and the Hecke-equivariance in Corollary 3.6). The base-change identification of characteristic ideals follows from the fact that the congruence map on zeta elements induces a pseudo-isomorphism of the associated Iwasawa modules whose kernel and cokernel are finite of order coprime to p (by the mod p non-vanishing hypothesis). Consequently no additional p-torsion is created. We nevertheless agree that an explicit verification of the ideal equality after base change would strengthen the exposition. We will insert a short auxiliary lemma immediately after the statement of Theorem 4.1 that computes the characteristic ideals under the relevant base change and confirms the absence of extra p-torsion. revision: yes

  2. Referee: [§5.2, Proposition 5.3] §5.2, Proposition 5.3: the claim that the μ-invariant vanishes for the propagated module when it vanishes for the base form uses the non-vanishing hypothesis only at the level of the mod p reduction; it is unclear whether the argument controls the higher p-power torsion that could appear in the Iwasawa module for weight >2.

    Authors: The vanishing of the μ-invariant for the propagated module is deduced from the equality of characteristic ideals (up to units) between the weight-2 and higher-weight cases, which is already established in Theorem 4.1. Because the characteristic ideal determines the μ-invariant independently of weight, and the mod p non-vanishing prevents p-torsion from appearing in the first place, higher p-power torsion cannot arise in the higher-weight module. The argument therefore does control the p-adic valuation. To make this control fully explicit we will add a paragraph in the proof of Proposition 5.3 that recalls the relation between the characteristic ideal and the μ-invariant and verifies that the base-change map preserves the p-adic valuation of the generator. revision: yes

Circularity Check

0 steps flagged

No significant circularity; result is conditional on explicit external assumption

full rationale

The paper's central claim is the propagation of Kato's main conjecture for higher-weight modular forms at arbitrary good primes, explicitly conditioned on the mod p non-vanishing of Kato's zeta elements. This non-vanishing is presented as an input hypothesis rather than a derived claim. The argument generalizes prior congruence results on Iwasawa invariants (Greenberg-Vatsal, Emerton-Pollack-Weston, etc.) without reducing any prediction or uniqueness statement to a self-citation chain or fitted parameter. No equations or steps in the provided abstract and structure exhibit self-definitional reduction, fitted-input-as-prediction, or ansatz smuggling. The derivation remains self-contained against the stated boundary condition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is present in the abstract.

pith-pipeline@v0.9.0 · 5648 in / 886 out tokens · 25003 ms · 2026-05-24T15:22:48.918231+00:00 · methodology

discussion (0)

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Reference graph

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