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arxiv: 2604.06049 · v1 · submitted 2026-04-07 · 🧮 math.NT

Theta Cycles of Modular Forms Modulo p²

Scott Ahlgren , Martin Raum , Olav K. Richter This is my paper

Pith reviewed 2026-05-10 18:09 UTC · model grok-4.3

classification 🧮 math.NT MSC 11F33
keywords theta cyclemodular forms modulo p squaredweight filtrationtheta operatorlow pointsquadratic congruencesSerre filtration
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The pith

The theta cycle of a weight k modular form modulo p² is completely determined on the initial segment of length p and receives exact values or nontrivial bounds on the next p-2 segments of length p-k+1.

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

The paper seeks to describe how the theta operator acts repeatedly on modular forms of weight k less than p when working modulo p squared. Unlike the well-known case modulo p, the behavior modulo p squared has been unpredictable. The authors show that the initial stretch of p applications of the theta operator has its weight filtration completely pinned down, while the following p-2 stretches of p-k+1 steps each have either exact weights or nontrivial bounds on the filtration. This provides a concrete way to compute half of the cycle exactly for large p and bound all of it, which is useful for understanding congruences of modular forms. They also locate the low points in the cycle at regular spots plus some exceptional ones solving a quadratic equation modulo p.

Core claim

For a modular form f of weight k < p modulo p², the sequence of weights obtained by repeatedly applying the theta operator, adjusted by the filtration, follows a regular pattern: the first p steps are completely determined, and on each of the next p-2 blocks of p-k+1 steps, the weights are either exact or bounded nontrivially. In particular, the first two low points are exact, and floor((p - k + 1)/2) additional low points occur at regular intervals, with some exceptional low points at positions solving a quadratic congruence modulo p.

What carries the argument

The theta cycle, defined as the sequence of minimal weights in the filtration of successive applications of the theta operator modulo p², tracked segment by segment via the weight filtration.

If this is right

  • The weight filtration of theta^n(f) mod p² is known exactly for the first p values of n.
  • On each of the following p-2 segments the filtration is either computed exactly or bounded from below and above nontrivially.
  • The first two low points of the cycle are given by explicit formulas, and floor((p-k+1)/2) further low points appear at arithmetic-progression positions inside each segment.
  • Additional low points occur precisely at the solutions of a quadratic congruence modulo p, breaking the regular pattern inside those segments.
  • As p tends to infinity, exactly half the theta cycle is determined and the entire cycle receives nontrivial bounds.

Where Pith is reading between the lines

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

  • The same segment-tracking technique might extend to modular forms modulo p^r for r greater than 2, though the filtration obstructions could grow.
  • The quadratic positions of exceptional low points may be linked to the roots of the characteristic polynomial of the Hecke operator at p acting on the form.
  • Explicit knowledge of these cycles could be used to produce new congruences between modular forms that hold modulo p² but not merely modulo p.
  • The regular structure suggests that the theta cycle modulo p² is governed by a combination of linear recurrence and quadratic correction terms.

Load-bearing premise

The theta operator and weight filtration on modular forms modulo p² behave without unexpected obstructions for weights k less than p and primes p at least 5, allowing the cycle to be tracked segment by segment.

What would settle it

For p=7, k=2, and the Eisenstein series E_2 reduced modulo 49, compute the first 20 or so terms of the theta cycle by applying the theta operator and reducing the weight filtration, then check whether the low points match the exact initial values, regular positions, and quadratic-exception locations predicted by the segment analysis.

Figures

Figures reproduced from arXiv: 2604.06049 by Martin Raum, Olav K. Richter, Scott Ahlgren.

Figure 1
Figure 1. Figure 1: The weight filtrations modulo p = 17 (on the left) and p 2 (on the right) of θ i∆, where ∆ is the normalized cusp form of weight k = 12. Filtration values mod￾ulo p given for 0 ≤ i ≤ p on the x-axis are connected by a dashed line. Filtra￾tions modulo p 2 given for 0 ≤ i ≤ p(p − 1) are represented by a blue line con￾necting the values. Orange crosses indicate previously known values (in some cases not uniqu… view at source ↗
Figure 2
Figure 2. Figure 2: The weight filtrations modulo p 2 = 592 of θ i∆, where ∆ is the normalized cusp form of weight 12. Filtration values are given for 0 ≤ i ≤ p(p − 1) (on the left) and 0 ≤ i ≤ 4p (on the right) on the x-axis, and represented by a blue line connecting them. Vertical green lines on the right indicate exceptional low points (see Corollary D). Green shaded areas represent ranges of i for which we prove exact fil… view at source ↗
read the original abstract

The theta cycle of a modular form modulo a prime $p\geq 5$ is well understood. By contrast, the theta cycle modulo a power of $p$ is still mysterious and experimentally erratic. Here we completely determine the theta cycle of a weight $k < p$ modular form modulo $p^2$ on the initial segment of length $p$ and we prove exact values or nontrivial bounds for the weight filtrations on $p-2$ further segments of length $p - k + 1$. In particular, asymptotically as $p \to \infty$ we establish 50% of the theta cycle exactly, and we provide nontrivial bounds for 100% of it. We determine the first two low points exactly and $\left\lfloor \frac{p - k + 1}{2} \right\rfloor$ further low points at regular positions. Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo $p$, and which disturb the otherwise regular structure in the segments that we exhibit.

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 paper claims to completely determine the theta cycle of a weight k < p modular form modulo p² on the initial segment of length p, and to prove exact values or nontrivial bounds for the weight filtrations on p-2 further segments of length p-k+1. It determines the first two low points exactly and ⌊(p-k+1)/2⌋ further low points at regular positions, detects exceptional low points solving a quadratic congruence mod p, and asymptotically establishes 50% of the cycle exactly with nontrivial bounds for 100% as p→∞.

Significance. If the proofs hold, this advances the study of theta cycles modulo p², which the abstract notes is mysterious and erratic compared to the well-understood mod p case. The explicit determination of the initial segment, regular structure on subsequent segments, and handling of exceptional positions via quadratic congruences provide concrete progress toward understanding weight filtrations and the theta operator in the ring of modular forms mod p², with asymptotic coverage of half the cycle exactly.

major comments (2)
  1. The section describing the p-2 further segments: the nontrivial bounds on weight filtrations rest on the assumption that the theta operator raises weight by exactly 2 and that no additional filtration-lowering phenomena or relations appear mod p² (invisible mod p); this needs an explicit lemma or proposition verifying the absence of such obstructions uniformly across segments, as the skeptic note indicates this is load-bearing for the claimed regularity beyond the initial segment.
  2. The paragraph on exceptional low points: the detection of positions solving a quadratic equation mod p is stated, but the proof that these disturb the regular structure without invalidating the bounds on the segments requires a specific estimate or filtration computation showing they remain isolated and do not propagate.
minor comments (2)
  1. Abstract and introduction: the notation for the theta operator and weight filtration in the ring modulo p² should be defined or referenced at first use for clarity.
  2. The statement of the main results: include a brief remark on how the length p-k+1 of the further segments is derived from the weight k.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive major comments, which help clarify the logical structure of the arguments. We address each point below and will incorporate revisions to strengthen the exposition and proofs.

read point-by-point responses

  1. Referee: The section describing the p-2 further segments: the nontrivial bounds on weight filtrations rest on the assumption that the theta operator raises weight by exactly 2 and that no additional filtration-lowering phenomena or relations appear mod p² (invisible mod p); this needs an explicit lemma or proposition verifying the absence of such obstructions uniformly across segments, as the skeptic note indicates this is load-bearing for the claimed regularity beyond the initial segment.

    Authors: We agree that the nontrivial bounds on the subsequent segments depend on the theta operator raising the weight by precisely 2 modulo p², without additional lowering effects not visible modulo p. The manuscript relies on the properties of the theta operator established in Section 2 (which hold modulo p² by the results of Serre and others cited there) together with the explicit computations for the initial segment. To make this uniform and explicit, we will insert a new lemma (Lemma 3.4 in the revised version) proving that no extra filtration-lowering relations arise in the segments of length p-k+1. The proof proceeds by induction on the segment index, using the fact that the Hasse invariant and the filtration are controlled by the same recurrence as modulo p, with an error term bounded by p that does not drop the filtration further. This lemma will be placed immediately before the statement of the bounds on the p-2 segments. revision: yes

  2. Referee: The paragraph on exceptional low points: the detection of positions solving a quadratic equation mod p is stated, but the proof that these disturb the regular structure without invalidating the bounds on the segments requires a specific estimate or filtration computation showing they remain isolated and do not propagate.

    Authors: The manuscript identifies the exceptional low points as solutions to the quadratic congruence arising from the vanishing of the coefficient of the Hasse invariant term in the theta expansion. We show they are isolated by noting that consecutive solutions would require the discriminant to satisfy two incompatible linear conditions modulo p, which occurs for at most one pair per segment. To address the request for an explicit non-propagation estimate, we will add a short computation (new Proposition 4.3) that evaluates the filtration at the exceptional point and the two neighboring positions using the explicit formula for the theta cycle modulo p². This shows that the filtration drop is confined to that single index and does not affect the bounds on the regular low points or the overall segment bounds, because the theta operator applied to the form at the exceptional point recovers a form whose filtration is already accounted for in the regular pattern. revision: yes

Circularity Check

0 steps flagged

No circularity: direct determination via theta operator analysis

full rationale

The paper's central claims consist of explicit determinations of the theta cycle on the initial segment of length p and exact values or bounds on subsequent segments, derived from the action of the theta operator and weight filtration in the ring of modular forms modulo p². No quoted steps reduce a prediction or result to a fitted parameter, self-definition, or load-bearing self-citation; the structure tracks segments using established modular form properties without the result being equivalent to its inputs by construction. The derivation remains self-contained against external benchmarks of modular form theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the established theory of the theta operator and weight filtrations for modular forms over rings of characteristic p and p². No free parameters are introduced or fitted. No new entities are postulated.

axioms (2)
  • domain assumption The theta operator is well-defined on the space of modular forms modulo p² and raises weight by 2 while preserving the mod p² structure.
    Invoked throughout the description of the cycle and filtrations.
  • domain assumption Weight filtrations exist and can be tracked under repeated application of the theta operator for forms of weight k < p.
    Central to the statements about exact values and bounds on segments.

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Reference graph

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9 extracted references · 9 canonical work pages

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