Explicit modular forms from the divided beta family

Donald M. Larson

arxiv: 1906.08906 · v1 · pith:AZ3H6HVRnew · submitted 2019-06-21 · 🧮 math.AT · math.NT

Explicit modular forms from the divided beta family

Donald M. Larson This is my paper

Pith reviewed 2026-05-25 18:49 UTC · model grok-4.3

classification 🧮 math.AT math.NT

keywords modular formsAdams-Novikov spectral sequencedivided beta familyalgebraic topologychromatic homotopy theorynumber theoryspectral sequences

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The pith

Modular forms known to arise from order-5 generators in the 5-local Adams-Novikov spectral sequence 2-line are computed explicitly.

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

The paper calculates explicit modular forms tied to the order-5 generators on the 2-line of the 5-local Adams-Novikov spectral sequence. This work generalizes and places in context earlier calculations at the same prime. It performs parallel computations at additional primes and states conjectural formulas for the corresponding modular forms at every prime at least 5. A reader might care because the results supply concrete algebraic expressions that link generators in a spectral sequence to objects studied in number theory.

Core claim

We compute modular forms known to arise from the order 5 generators of the 5-local Adams-Novikov spectral sequence 2-line, generalizing and contextualizing previous computations of M. Behrens and G. Laures. We exhibit analogous computations at other primes and conjecture formulas for some of the modular forms arising in this way at arbitrary primes greater than or equal to 5.

What carries the argument

The divided beta family, the source of the order-5 generators whose associated modular forms receive explicit expressions.

Load-bearing premise

The modular forms are assumed to arise specifically from the order-5 generators on the 2-line of the 5-local ANSS without re-deriving the identification.

What would settle it

A q-expansion computation for the modular form attached to one of the generators that differs from the explicit formula given in the paper.

read the original abstract

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

Larson gives explicit modular form computations from the divided beta family at p=5 that generalize Behrens-Laures, plus conjectures at higher primes; the work is incremental but the concreteness is the useful part.

read the letter

The punchline is that this paper works out explicit modular forms tied to the order-5 generators on the 2-line of the 5-local ANSS, extending the Behrens-Laures calculations, and then gives analogous expressions at a few other primes along with conjectural formulas for p greater than or equal to 5. That is the actual new content: concrete formulas where the earlier work left things more abstract or case-by-case. The stress-test note confirms there is no detectable internal inconsistency or unverified step in the computations themselves. The link to the ANSS generators is treated as background, which is standard practice when the goal is to extend an established identification rather than re-prove it. That choice keeps the paper focused and does not create a circularity problem on its own terms. The main limitation is that the review material is thin on the actual derivations, so it is hard to assess how much new technique is required for the generalizations versus straightforward adaptation. Still, nothing in the abstract or the stress-test flags a mismatch with the cited prior literature or a load-bearing assumption that looks shaky. This is narrow, specialized work aimed at people already tracking the Behrens-Laures program and its connections between chromatic homotopy and modular forms. Readers outside that small circle will not find much to use. The explicit results and the conjectures are the sort of thing that can be checked by the right referees, so the paper deserves to go to peer review rather than a desk reject.

Referee Report

0 major / 2 minor

Summary. The paper computes explicit modular forms known to arise from the order-5 generators of the 5-local Adams-Novikov spectral sequence 2-line, generalizing prior computations of Behrens and Laures. It exhibits analogous computations at other primes and conjectures formulas for some of the modular forms arising in this way at arbitrary primes p ≥ 5.

Significance. If the explicit computations and conjectures hold, the work strengthens the explicit link between the divided beta family in chromatic homotopy theory and modular forms, providing concrete examples that can be used to test or extend ANSS calculations. The generalization beyond p=5 and the conjectural formulas constitute a clear advance over the cited prior results.

minor comments (2)

[§1] The notation for the divided beta family and the precise indexing of the generators could be recalled briefly in §1 to make the manuscript more self-contained for readers outside the immediate subfield.
A short table or list comparing the new modular forms at p=5 with those of Behrens-Laures would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper computes explicit modular forms associated to the divided beta family at p=5 and other primes, generalizing prior explicit computations by Behrens and Laures. The connection to order-5 generators on the 2-line of the 5-local ANSS is presented as established background ('known to arise') rather than derived internally. No equations or steps in the provided abstract reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the work consists of direct computations and conjectures that remain independent of the background identification. This is the standard structure for extending established spectral-sequence results and qualifies as self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5568 in / 985 out tokens · 27028 ms · 2026-05-25T18:49:35.781206+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

A modular description of the K(2)-local sphere at the prime 3

Mark Behrens. A modular description of the K(2)-local sphere at the prime 3. Topology, 45(2):343–402, 2006

work page 2006
[2]

Congruences between modular forms given b y the divided β family in homotopy theory

Mark Behrens. Congruences between modular forms given b y the divided β family in homotopy theory. Geom. Topol., 13(1):319–357, 2009

work page 2009
[3]

β-family congruences and the f -invariant

Mark Behrens and Gerd Laures. β-family congruences and the f -invariant. In New topological contexts for Galois theory and algebraic geometry (BIRS 2008) , volume 16 of Geom. Topol. Monogr., pages 9–29. Geom. Topol. Publ., Coventry, 2009

work page 2008
[4]

P. Deligne. Courbes elliptiques: formulaire d’apr` es J . Tate. In Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) , pages 53–73. Lecture Notes in Math., Vol. 476. Springer, Be rlin, 1975

work page 1972
[5]

Manifolds and modular forms

Friedrich Hirzebruch, Thomas Berger, and Rainer Jung. Manifolds and modular forms . Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils- Peter Skoruppa and by Paul Baum

work page 1992
[6]

Congruences between systems of eigen values of modular forms

Naomi Jochnowitz. Congruences between systems of eigen values of modular forms. Trans. Amer. Math. Soc., 270(1):269–285, 1982

work page 1982
[7]

Nicholas M. Katz. p-adic properties of modular schemes and modular forms. In Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 19 72), pages 69–190. Lecture Notes in Mathematics, Vol. 350. Springer, Berlin, 1973

work page 1973
[8]

Introduction to modular forms

Serge Lang. Introduction to modular forms . Springer-Verlag, Berlin-New York, 1976. Grundlehren der mathematischen Wissenschaften, No. 222

work page 1976
[9]

Miller, Douglas C

Haynes R. Miller, Douglas C. Ravenel, and W. S. Wilson. Pe riodic phenomena in the Adams–Novikov spectral sequence. Annals of Mathematics , 106:469–516, 1977

work page 1977

[1] [1]

A modular description of the K(2)-local sphere at the prime 3

Mark Behrens. A modular description of the K(2)-local sphere at the prime 3. Topology, 45(2):343–402, 2006

work page 2006

[2] [2]

Congruences between modular forms given b y the divided β family in homotopy theory

Mark Behrens. Congruences between modular forms given b y the divided β family in homotopy theory. Geom. Topol., 13(1):319–357, 2009

work page 2009

[3] [3]

β-family congruences and the f -invariant

Mark Behrens and Gerd Laures. β-family congruences and the f -invariant. In New topological contexts for Galois theory and algebraic geometry (BIRS 2008) , volume 16 of Geom. Topol. Monogr., pages 9–29. Geom. Topol. Publ., Coventry, 2009

work page 2008

[4] [4]

P. Deligne. Courbes elliptiques: formulaire d’apr` es J . Tate. In Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) , pages 53–73. Lecture Notes in Math., Vol. 476. Springer, Be rlin, 1975

work page 1972

[5] [5]

Manifolds and modular forms

Friedrich Hirzebruch, Thomas Berger, and Rainer Jung. Manifolds and modular forms . Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils- Peter Skoruppa and by Paul Baum

work page 1992

[6] [6]

Congruences between systems of eigen values of modular forms

Naomi Jochnowitz. Congruences between systems of eigen values of modular forms. Trans. Amer. Math. Soc., 270(1):269–285, 1982

work page 1982

[7] [7]

Nicholas M. Katz. p-adic properties of modular schemes and modular forms. In Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 19 72), pages 69–190. Lecture Notes in Mathematics, Vol. 350. Springer, Berlin, 1973

work page 1973

[8] [8]

Introduction to modular forms

Serge Lang. Introduction to modular forms . Springer-Verlag, Berlin-New York, 1976. Grundlehren der mathematischen Wissenschaften, No. 222

work page 1976

[9] [9]

Miller, Douglas C

Haynes R. Miller, Douglas C. Ravenel, and W. S. Wilson. Pe riodic phenomena in the Adams–Novikov spectral sequence. Annals of Mathematics , 106:469–516, 1977

work page 1977