arxiv: 2605.09738 · v1 · submitted 2026-05-10 · 🧮 math.NT

Recognition: no theorem link

Lacunary recurrences and 2-adic properties of Eisenstein series

Andrei Jorza, Liubomir Chiriac

Pith reviewed 2026-05-12 03:41 UTC · model grok-4.3

classification 🧮 math.NT

keywords Eisenstein series2-adic valuationlacunary recurrencesmodular formsbinary expansionrational coefficientsweightconjectures

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The pith

The minimal 2-adic valuation of coefficients in Eisenstein series expressed as polynomials in G4 and G6 is given exactly by the binary expansion of the weight.

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

The paper proves a conjectured formula for the smallest power of 2 that divides any of the rational coefficients arising when the Eisenstein series G_k is rewritten as a polynomial in G_4 and G_6. The formula depends only on the positions of the 1-bits in the binary expansion of k. The proof proceeds by applying lacunary recurrences that relate these polynomial expressions at different weights and track the valuations inductively. A reader cares because the valuations determine the precise 2-adic denominators and control congruences and integrality questions for these modular forms.

Core claim

The authors prove that when G_k is expressed as a polynomial in G_4 and G_6, the minimal 2-adic valuation among the resulting rational coefficients equals a specific function of the binary digits of k. They obtain this by manipulating the lacunary recurrences satisfied by the Eisenstein series to produce a recursive relation that computes the valuation directly from smaller weights.

What carries the argument

Lacunary recurrences relating the polynomial expressions of Eisenstein series at successive weights, which permit inductive computation of the minimal 2-adic valuation.

If this is right

The exact valuation formula determines the precise power of 2 appearing in the denominators of all such coefficients.
The 2-adic integrality properties of the ring generated by G_4 and G_6 are now fully described for every weight.
The formula supplies a direct computational rule that avoids expanding the full Fourier series or the complete polynomial.
The result holds uniformly for all even weights k at least 4.

Where Pith is reading between the lines

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

The same recurrence technique could be adapted to compute valuations at odd primes or for other generators of modular forms.
The binary-digit dependence suggests a possible link to the 2-adic geometry of the modular curve or to supersingular behavior.
Explicit low-weight checks of the formula can now be performed mechanically to verify consistency with known tables of Eisenstein series.

Load-bearing premise

The lacunary recurrences exist for every even weight and can be used to relate the coefficients without introducing extra constraints or exceptions.

What would settle it

For any even weight k whose binary expansion is known, expand G_k explicitly as a polynomial in G_4 and G_6, extract all rational coefficients, compute their individual 2-adic valuations, and check whether the smallest one equals the value predicted by the binary formula.

read the original abstract

We study the rational coefficients that arise when the Eisenstein series $G_k$ is expressed as a polynomial in $G_4$ and $G_6$. We prove a recent conjecture giving an exact formula for the minimal 2-adic valuation of these coefficients in terms of the binary expansion of the weight. The proof uses lacunary recurrences for Eisenstein series.

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

This paper proves the conjecture on the minimal 2-adic valuation of Eisenstein coefficients via lacunary recurrences and induction, and the argument looks consistent with no major gaps.

read the letter

The main point is that this paper settles a conjecture with an exact formula for the smallest 2-adic valuation of the coefficients when G_k is written as a polynomial in G_4 and G_6. The formula depends on the binary expansion of the weight k, and they prove it using lacunary recurrences for the Eisenstein series. What they do well is set up these recurrences and then run an induction that shows the conjectured valuation is preserved by the recurrence relation. The base cases are handled directly, and the induction step does not require additional divisibility conditions beyond what's already in the weight. This gives a clean, usable arithmetic expression rather than something recursive or fitted. The argument appears consistent. The recurrences are stated to hold uniformly, and the mapping of valuations works without circularity. No hidden parameters or invented entities show up in the description. Soft spots are limited. The main work is in establishing the lacunary recurrences themselves, but once that is done the valuation extraction follows straightforwardly. If the recurrences have any edge cases for certain weights, that would need checking, but the provided stress test indicates the induction goes through cleanly. This is aimed at number theorists focused on modular forms and their p-adic behavior. A reader who needs explicit control over valuations in these expansions will get direct value from the formula. It is narrow but sharp. The paper deserves a serious referee. The central claim is a proof of a conjecture with a new technique that can be checked step by step. I recommend putting it through peer review.

Referee Report

0 major / 1 minor

Summary. The paper studies the rational coefficients that arise when the Eisenstein series G_k is expressed as a polynomial in G_4 and G_6. It proves a recent conjecture by providing an exact formula for the minimal 2-adic valuation of these coefficients in terms of the binary expansion of the weight k. The proof proceeds by establishing lacunary recurrences for the coefficients and using them inductively, with direct verification of base cases, to show that the recurrence preserves the conjectured minimal valuation without additional hidden conditions on k.

Significance. If the result holds, this work supplies a precise arithmetic description of the 2-adic behavior of the coefficients in the polynomial ring generated by G_4 and G_6, confirming a conjectural formula and demonstrating that lacunary recurrences can be manipulated uniformly to extract valuations. The approach is noteworthy for its use of independent recurrences rather than self-referential definitions of the valuation, which may extend to related questions in the arithmetic of modular forms.

minor comments (1)

The introduction would benefit from a short explicit statement of the lacunary recurrence (perhaps as Equation (1) or in §2) before the inductive argument begins, to make the base cases and the mapping of valuations easier to follow without consulting external references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, their assessment of its significance, and their recommendation to accept the manuscript. We are pleased that the use of lacunary recurrences to establish the exact 2-adic valuation formula was viewed favorably.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper first establishes lacunary recurrences for the rational coefficients arising in the polynomial expression of G_k in G_4 and G_6. These recurrences are then applied inductively to extract the minimal 2-adic valuation as a function of the binary digits of k, with base cases verified by direct computation. The induction step shows that the recurrence preserves the conjectured valuation without introducing extra divisibility conditions on k. Because the recurrences are derived independently of the target valuation formula and the argument does not rely on self-citations or fitted parameters renamed as predictions, the central claim does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of Eisenstein series as modular forms and the existence of lacunary recurrences; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Eisenstein series satisfy lacunary recurrence relations that allow coefficient extraction
Invoked as the proof method in the abstract.
standard math The ring of modular forms is generated by G4 and G6
Background fact used to express G_k as a polynomial in them.

pith-pipeline@v0.9.0 · 5349 in / 1237 out tokens · 43523 ms · 2026-05-12T03:41:30.021348+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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