arxiv: 2605.04195 · v1 · submitted 2026-05-05 · 🧮 math.NT

Recognition: 3 theorem links

· Lean Theorem

A new perspective on the rank of Mazur's Eisenstein Hecke algebra

Jaclyn Lang , Katharina M\"uller , Bharathwaj Palvannan

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:44 UTC · model grok-4.3

classification 🧮 math.NT

keywords Eisenstein Hecke algebraMazurrankzeta elementDirichlet L-valuesmodular formsIwasawa algebraGalois orbits

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The pith

The rank r of the Eisenstein Hecke algebra equals one plus the vanishing order of the mod-p zeta element when r is 2 or 3.

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

The paper studies the rank r of the Hecke algebra that parametrizes weight-2 modular forms of level N congruent to Eisenstein series modulo a prime p. It proves that when r equals 2 or 3, r minus 1 equals the order of vanishing at p of the reduction modulo p of a zeta element that interpolates Dirichlet L-values at s equals negative one. A sympathetic reader would care because this recovers earlier theorems of Merel and Lecouturier and supplies a uniform method that works for all ranks by moving to the Hecke algebra in level N squared. The work also identifies Galois orbits of cuspidal newforms in level N squared that are Eisenstein modulo p in the cases where exactly one of the two quantities equals 3.

Core claim

Let N and p be primes at least 5 with N congruent to 1 modulo p. Let r be the rank of the Hecke algebra parametrizing weight-2 level-N modular forms that are Eisenstein modulo p. When r equals 2 or 3, r minus 1 equals the order of vanishing of the mod-p reduction of the zeta element that interpolates Dirichlet L-values at negative one. The equality can fail when r is at least 4. The authors treat all cases uniformly by studying the analogous Hecke algebra in level N squared and give precise information about Galois orbits of cuspidal newforms in that level when exactly one of r minus 1 or the vanishing order equals 3.

What carries the argument

The rank r of the Eisenstein Hecke algebra in level N, linked to the vanishing order of the mod-p reduction of the zeta element that interpolates Dirichlet L-values at -1.

If this is right

Results of Merel and Lecouturier on the rank of the Eisenstein Hecke algebra are recovered as special cases.
The uniform study via the Hecke algebra in level N squared applies to all values of the rank.
When r minus 1 equals 3 but the vanishing order does not, or vice versa, the Galois orbits of the cuspidal newforms in level N squared that are Eisenstein modulo p can be described precisely.
The equality between r minus 1 and the vanishing order can fail when the rank reaches 4 or higher.

Where Pith is reading between the lines

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

The comparison between rank and vanishing order might extend to compute the rank in additional cases by using p-adic L-functions directly.
The uniform treatment at level N squared could apply to similar questions about Hecke algebras for other congruence classes or higher weights.
The heuristic explaining failure at higher ranks suggests that extra factors in the Iwasawa algebra may account for the discrepancy.

Load-bearing premise

The argument takes as given the existence and basic interpolation properties of the zeta element for Dirichlet L-values at -1 together with the identification of the Hecke algebra as a quotient of the Iwasawa algebra.

What would settle it

An explicit pair of primes N and p satisfying N congruent to 1 modulo p, with the computed rank r equal to 2 or 3, but where r minus 1 fails to equal the computed vanishing order of the mod-p zeta element, would disprove the central equality.

read the original abstract

Let $N, p \geq 5$ be primes such that $N \equiv 1 \bmod p$. We study the rank $r$ of the Hecke algebra that parametrizes modular forms of weight 2 and level $N$ that are Eisenstein modulo $p$. When $r$ is $2$ or $3$, we prove that $r-1$ equals the order of vanishing of the mod-$p$ reduction of a zeta element that interpolates Dirichlet $L$-values at $-1$, thereby recovering results of Merel and Lecouturier. This equality can fail in some cases when $r \geq 4$, and we provide a heuristic explanation of this failure. Our approach handles all of these cases uniformly by studying the analogous Hecke algebra in level $N^2$. When exactly one of $r-1$ or the order of vanishing equals $3$, we provide precise information about Galois orbits of cuspidal newforms in level $N^2$ that are Eisenstein modulo $p$.

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 unifies the rank of the Eisenstein Hecke algebra with the vanishing order of a mod-p zeta element via a level-N² lift, proving the link for r=2 and 3 while giving a heuristic for failures at higher rank.

read the letter

The main point is that the authors relate the rank r of the mod-p Eisenstein Hecke algebra in level N to the vanishing order of a zeta element that interpolates Dirichlet L-values at -1. For r equal to 2 or 3 they prove r minus 1 equals that order, recovering Merel and Lecouturier. They do this uniformly by working with the Hecke algebra in level N squared instead of handling small ranks separately, and they extract Galois orbit information on the cuspidal newforms when exactly one of the two quantities is 3. For r at least 4 they note that the equality can fail and supply a heuristic based on the same level-N² setup.

Referee Report

0 major / 3 minor

Summary. The paper studies the rank r of the Eisenstein Hecke algebra parametrizing weight-2 modular forms of level N that are Eisenstein modulo p, for primes N, p ≥ 5 with N ≡ 1 mod p. For r = 2 or 3 it proves that r−1 equals the p-adic order of vanishing of the mod-p reduction of a zeta element interpolating Dirichlet L-values at −1, recovering the theorems of Merel and Lecouturier. The argument proceeds uniformly by comparing the Hecke algebra at level N with the analogous algebra at level N²; for r ≥ 4 the equality is shown to fail in some cases and a heuristic explanation is supplied. When exactly one of r−1 or the vanishing order equals 3, the paper gives precise information on the Galois orbits of the corresponding cuspidal newforms in level N².

Significance. If the proofs are complete, the work supplies a uniform arithmetic perspective that recovers two classical results as special cases of a single comparison between Hecke-algebra rank and the vanishing of an Iwasawa-theoretic zeta element. The extension to higher rank with explicit counter-examples and a heuristic, together with the Galois-orbit statements, adds new structural information about Eisenstein congruences at level N². The approach relies on standard inputs (existence of the zeta element and the identification of the level-N Hecke algebra as a quotient of the Iwasawa algebra) and therefore strengthens the link between Mazur’s Eisenstein ideal and p-adic L-functions without introducing new ad-hoc constructions.

minor comments (3)

[§1] §1, paragraph following Theorem 1.1: the precise statement of the interpolation property of the zeta element (the precise set of characters and the exact normalization of the L-values) is only sketched; an explicit formula or reference to the normalization used in the later sections would improve readability.
[§5] §5, discussion of the heuristic for r ≥ 4: the failure examples are listed but the numerical verification that the Hecke algebra rank is indeed r in those cases is not tabulated; adding a short table of the computed ranks and the corresponding vanishing orders would make the heuristic more concrete.
[Notation] Notation: the symbol T_N for the Hecke algebra at level N is introduced without an explicit definition of its Eisenstein quotient; a one-sentence reminder of the precise quotient (e.g., by the Eisenstein ideal) at the first occurrence would prevent minor confusion for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. The referee's description correctly identifies the main contributions: the uniform comparison between the Hecke algebras at levels N and N², the recovery of the Merel–Lecouturier theorems for ranks 2 and 3 via the vanishing order of the mod-p zeta element, the explicit counter-examples and heuristic for r ≥ 4, and the Galois-orbit statements when one of the quantities equals 3. We are pleased that the work is viewed as strengthening the link between Mazur’s Eisenstein ideal and p-adic L-functions using only standard inputs.

Circularity Check

0 steps flagged

No significant circularity; central equality uses independent prior inputs

full rationale

The paper proves that for r=2 or 3 the Hecke-algebra rank r minus one equals the vanishing order of the mod-p zeta element (recovering Merel-Lecouturier) by studying the level-N² Eisenstein Hecke algebra uniformly. The existence and interpolation properties of the zeta element, together with the identification of the level-N Hecke algebra as a quotient of the Iwasawa algebra, are explicitly taken as given from prior literature. No equation or step in the derivation reduces the target equality to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the cited results of Merel and Lecouturier are external and the new argument supplies independent content for the r=2,3 cases.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard properties of Hecke algebras, modular forms of weight 2, and the existence of a zeta element interpolating Dirichlet L-values at -1; no free parameters, invented entities, or ad-hoc axioms are explicitly introduced in the summary.

axioms (2)

domain assumption Existence of a zeta element in the Iwasawa algebra that interpolates Dirichlet L-values at s = -1 and whose mod-p reduction has a well-defined order of vanishing.
Invoked in the statement relating the vanishing order to the Hecke-algebra rank.
domain assumption The Hecke algebra in level N that parametrizes Eisenstein forms modulo p is a quotient of the Iwasawa algebra in a manner compatible with the zeta element.
Required for the equality r-1 = vanishing order to make sense.

pith-pipeline@v0.9.0 · 5491 in / 1744 out tokens · 36626 ms · 2026-05-08T17:44:21.154824+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J(x)=½(x+x⁻¹)-1 unique calibrated cost) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Let ζ := -N/2 · Σ B_2(i/N)[i+NZ] ∈ Z_p[(Z/NZ)^×] ... Theorem A: r=2 iff ord(ζ̄)=1; Theorem B: r=3 iff ord(ζ̄)=2.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat initiality / Peano recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The ring T is a Z_p[Δ]^+-algebra, and it is free of rank r as a Z_p[Δ]^+-module. Hence rk_{Z_p} T = ½(p^s + 1)r.
IndisputableMonolith/Constants and IndisputableMonolith/Foundation/AlexanderDuality.lean phi_golden_ratio; alexander_duality_circle_linking (D=3) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The whole paper develops Hecke algebras, modular forms of weight 2, deformation rings, residue exact sequences on modular curves; no cost function, no golden ratio, no 8-tick clock, no derivation of physical constants.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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