Quasiautomorphic forms are isomorphic to vector-valued automorphic forms

Michael Andrew Henry

arxiv: 2605.21067 · v1 · pith:A5QZ6UH3new · submitted 2026-05-20 · 🧮 math.NT

Quasiautomorphic forms are isomorphic to vector-valued automorphic forms

Michael Andrew Henry This is my paper

Pith reviewed 2026-05-21 02:01 UTC · model grok-4.3

classification 🧮 math.NT

keywords quasiautomorphic formsHecke vector-formsvector-valued automorphic formsHecke triangle groupsmultiplier systemsquasimodular formsbijectionfunctional equations

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The pith

Quasiautomorphic forms over Hecke triangle groups correspond one-to-one with Hecke vector-forms.

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

The paper proves that quasiautomorphic forms defined with respect to a Hecke triangle group stand in bijection with a certain class of vector-valued automorphic forms called Hecke vector-forms. It constructs the forward map that turns each quasiautomorphic form into a vector of functions, then verifies that this vector obeys the transformation laws required of automorphic forms when the group generators act. The inverse map is built by using the multiplier system of the vector form to assemble the original quasiautomorphic form, showing that every such vector form comes from exactly one quasiautomorphic form. Because the ordinary modular group is a Hecke triangle group, the same bijection applies directly to quasimodular forms. This means that any result proved for one collection of objects immediately yields a parallel result for the other collection.

Core claim

The central claim is that a natural bijection exists between the space of quasiautomorphic forms over a Hecke triangle group and the space of Hecke vector-forms. The map is defined by sending a quasiautomorphic form to a vector whose entries are suitable transformations of it. Functional equations for the vector form are established by checking them on the generators of the group. The inverse is obtained by applying the multiplier system of the Hecke vector-form to recover the quasiautomorphic form.

What carries the argument

The bijection constructed by mapping a quasiautomorphic form to its associated Hecke vector-form and recovering it via the multiplier system.

Load-bearing premise

The multiplier system of the Hecke vector-forms must be compatible with the Hecke triangle group action in a way that makes the inverse map well-defined and exactly recovers each original quasiautomorphic form.

What would settle it

The claim would be falsified by exhibiting a quasiautomorphic form for which the constructed vector fails to satisfy one of the required functional equations, or by finding a Hecke vector-form that does not arise as the image of any quasiautomorphic form under the proposed map.

Figures

Figures reproduced from arXiv: 2605.21067 by Michael Andrew Henry.

**Figure 2.** Figure 2: Locally, the normalized initial triangle [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

We utilize the structure of quasiautomorphic forms over a Hecke triangle group to define a mapping from a quasiautomorphic form to a vector-valued automorphic form (vvaf). This kind of vvaf we call a Hecke vector-form. First we supply a proof of the functional equations that hold for Hecke vector-forms modulo the group generators. Then, utilizing the multiplier system for these Hecke vector-forms, we prove the opposite direction and complete the bijection. Since the modular group is a special instance of the Hecke triangle groups, our results hold for quasimodular forms.

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 gives an explicit bijection from quasiautomorphic forms on Hecke triangle groups to a class of vector-valued forms, but the argument only checks the functional equations on generators and leaves the compatibility of the multiplier system with the full group relations unaddressed.

read the letter

The main takeaway is that this paper constructs a bijection between quasiautomorphic forms over general Hecke triangle groups and what it calls Hecke vector-forms. It extends the known modular-group case and supplies a forward map plus an inverse built from the multiplier system. That is the concrete new piece: a translation between two descriptions that had been treated separately in this corner of the theory. The modular-group specialization then recovers the quasimodular case as a corollary, which is a clean way to organize existing results. The outline is direct—define the map, verify the equations on the generators, then invert—so the logic is easy to follow on paper. If the details hold, it could serve as a practical dictionary for calculations that mix the two languages. The soft spot is exactly the one flagged in the stress-test. The functional equations are established only modulo the generators, and the inverse relies on the multiplier system being compatible with the full set of relations in the group presentation. Nothing in the abstract or the high-level sketch shows an independent check that the constructed multiplier respects those relations (for instance S² = 1 or the cubic relation). Without that step the map may not descend to a well-defined object on the quotient, which would make the claimed bijection incomplete. The paper does not appear to contain circular definitions or fitted parameters, and the argument proceeds outward from the group action rather than inward from a self-referential claim. Still, the gap on relations is load-bearing for the isomorphism. This is narrow work aimed at people already working with automorphic forms on triangle groups or with quasimodular forms. A reader who needs to move forms between scalar and vector-valued settings might get a usable tool out of it once the compatibility is settled. The claim is specific enough and the outline coherent enough that it should go to a serious referee rather than a desk reject; the referee can check whether the multiplier construction actually closes the loop on the relations. I would send it for review.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes a bijection between the space of quasiautomorphic forms on a Hecke triangle group and the space of Hecke vector-valued automorphic forms (vvafs). The forward direction defines a map from quasiautomorphic forms to these vector-forms and verifies that it satisfies the defining functional equations when checked against the generators of the group. The reverse direction uses the associated multiplier system to construct an inverse map, completing the isomorphism. The result specializes to the case of quasimodular forms when the group is the modular group.

Significance. If the bijection holds with the required compatibility, the result would supply a concrete structural correspondence between quasiautomorphic forms and a class of vector-valued forms on Hecke triangle groups. This could allow transfer of analytic or arithmetic techniques between the two settings and would extend known relations for the modular group to the broader family of Hecke triangle groups.

major comments (1)

[Construction of the inverse map and proof of the bijection] The forward map is verified to satisfy the functional equations only modulo the generators of the Hecke triangle group. For the inverse map constructed via the multiplier system to yield a genuine vector-valued automorphic form on the full group, the multiplier must be shown to be compatible with the complete set of relations in the group presentation (e.g., S² = 1 and (ST)³ = 1 for the modular-group case). No explicit verification of these relations for the constructed multiplier is supplied beyond the generators; this compatibility is load-bearing for the claim that the inverse recovers a well-defined automorphic form without ambiguity.

minor comments (2)

[Introduction and definitions] Notation for the multiplier system and the precise definition of a Hecke vector-form should be introduced with a dedicated paragraph or displayed equation early in the manuscript to improve readability.
[Final section] The specialization statement for the modular group would benefit from a short explicit corollary stating the resulting bijection for quasimodular forms.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive major comment. We address the point directly below and will incorporate an explicit verification to strengthen the presentation of the bijection.

read point-by-point responses

Referee: [Construction of the inverse map and proof of the bijection] The forward map is verified to satisfy the functional equations only modulo the group generators. For the inverse map constructed via the multiplier system to yield a genuine vector-valued automorphic form on the full group, the multiplier must be shown to be compatible with the complete set of relations in the group presentation (e.g., S² = 1 and (ST)³ = 1 for the modular-group case). No explicit verification of these relations for the constructed multiplier is supplied beyond the generators; this compatibility is load-bearing for the claim that the inverse recovers a well-defined automorphic form without ambiguity.

Authors: We agree that an explicit check of compatibility with the full set of relations in the group presentation is desirable for clarity. The manuscript verifies the functional equations on the generators (which generate the group) and constructs the inverse map using the multiplier system associated to the Hecke vector-form. By the definition of the multiplier system arising from the quasiautomorphic form, consistency under the relations is inherited from the transformation properties already established. Nevertheless, to address the referee’s concern directly, we will add a short subsection in the revised version that explicitly verifies the multiplier satisfies the defining relations of the Hecke triangle group (including the analogues of S² = 1 and (ST)³ = 1, as well as the corresponding relations for general Hecke triangle groups). This addition will confirm that the inverse map is unambiguously defined on the entire group. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from definitions and group generators to bijection

full rationale

The paper defines a forward map from quasiautomorphic forms to Hecke vector-forms, proves the functional equations hold modulo the generators of the Hecke triangle group, and then uses the multiplier system to construct the inverse map establishing the bijection. This chain relies on explicit verification of the defining properties and compatibility with the group action rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. The extension to the modular group as a special case follows directly from the general construction without reducing the central isomorphism to its inputs by construction. The argument is therefore self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard background facts from the theory of Hecke groups and automorphic forms; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Quasiautomorphic forms over a Hecke triangle group possess a well-defined structure that permits a mapping to vector-valued forms satisfying functional equations under the group generators.
Invoked at the outset when the mapping is defined.

pith-pipeline@v0.9.0 · 5617 in / 1338 out tokens · 53648 ms · 2026-05-21T02:01:33.914193+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Theorem 1 (Isomorphism theorem). The function U_{t_μ,w,r}(z)=U is a quasiautomorphic form ... if and only if there is a vector-valued automorphic form F_U(z) such that F_U(Tz)=ε_r(T)F_U(z), F_U(Sz)/z^{w-r}=ε_r(S)F_U(z) for generators T,S ... multiplier system ε_r ... (r+1)×(r+1) matrices.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

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

Theorem 4. ... ε_r : t_μ → GL_{r+1}(K) if ε_r(u) = exp((ζ+ζ^{-1})A_r) if u=T, d_y^r((-1)^j) if u=S. ... ε_r ≅ Sym^r ε_1

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

Works this paper leans on

33 extracted references · 33 canonical work pages

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A unified matrix approach to the representation of Appell polynomials

L. Aceto, H. R. Malonek, and G. Tomaz. “A unified matrix approach to the representation of Appell polynomials”. In:Integral Transforms Spec. Funct.26.6 (2015), pp. 426–441

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The matrices of Pascal and other greats

L. Aceto and D. Trigiante. “The matrices of Pascal and other greats”. In:Am. Math. Mon.108.3 (2001), pp. 232–245

work page 2001

[3] [3]

Aspects of Hecke symme- try: anomalies, curves, and Chazy equations

S. K. Ashok, D. P. Jatkar, and M. Raman. “Aspects of Hecke symme- try: anomalies, curves, and Chazy equations”. In:SIGMA Symmetry Integrability Geom. Methods Appl.16 (2020), Paper No. 001, 26

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Au- tomorphic forms for triangle groups

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Conformal field theory techniques in largeNYang- Mills theory

M. R. Douglas. “Conformal field theory techniques in largeNYang- Mills theory”. In:Quantum field theory and string theory. Proceed- ings of a NATO advanced research workshop on new developments in string theory, conformal models, and topological field theory, in Carg` ese, France, May 10-21, 1993. Plenum, New York, NY, 1995, pp. 119–135

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Pascal matrices

A. Edelman and G. Strang. “Pascal matrices”. In:Amer. Math. Monthly 111.3 (2004), pp. 189–197. 24

work page 2004

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M. A. Henry. “Introduction to Hecke vector-forms: a generalized au- tomorphic form as a vector over a Hecke triangle group”. PhD thesis. TU Graz, 2025

work page 2025

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R. A. Horn and C. R. Johnson.Matrix analysis. Cambridge University Press, Cambridge, MA, 1985

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work page 2021

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On extremal quasimodular forms

M. Kaneko and M. Koike. “On extremal quasimodular forms”. In: Kyushu J. Math.60.2 (2006), pp. 457–470

work page 2006

[19] [19]

A generalized Jacobi theta function and quasimodular forms

M. Kaneko and D. Zagier. “A generalized Jacobi theta function and quasimodular forms”. In:The moduli space of curves (Texel Island, 1994). Vol. 129. Progr. Math. Birkh¨ auser Boston, Boston, MA, 1995, pp. 165–172

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S. K. Lando and A. K. Zvonkin.Graphs on surfaces and their appli- cations. Appendix by Don B. Zagier. Springer, Berlin, 2004

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Lehner.Discontinuous groups and automorphic functions

J. Lehner.Discontinuous groups and automorphic functions. Vol. No. VIII. Mathematical Surveys. American Mathematical Society, Provi- dence, RI, 1964, pp. xi+425

work page 1964

[23] [23]

Perturbations, deformations, and variations (and “near- misses

B. Mazur. “Perturbations, deformations, and variations (and “near- misses”) in geometry, physics, and number theory”. In:Bull. Amer. Math. Soc. (N.S.)41.3 (2004), pp. 307–336

work page 2004

[24] [24]

Determination of normalized extremal quasimodular forms of depth 1 with integral Fourier coefficients

T. Nakaya. “Determination of normalized extremal quasimodular forms of depth 1 with integral Fourier coefficients”. In:Int. J. Number The- ory20.3 (2024), pp. 641–689

work page 2024

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Nehari.Conformal mapping

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work page 1952

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Counting hyperelliptic curves on an Abelian surface with quasi-modular forms

S. C. F. Rose. “Counting hyperelliptic curves on an Abelian surface with quasi-modular forms”. In:Commun. Number Theory Phys.8.2 (2014), pp. 243–293

work page 2014

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Roy.Elliptic and modular functions from Gauss to Dedekind to Hecke

R. Roy.Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge, 2017

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Arithmetic triangle groups

K. Takeuchi. “Arithmetic triangle groups”. In:J. Math. Soc. Japan29 (1977), pp. 91–106

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On the arithmetic of D-brane superpotentials. Lines and conics on the mirror quintic

J. Walcher. “On the arithmetic of D-brane superpotentials. Lines and conics on the mirror quintic”. In:Commun. Number Theory Phys.6.2 (2012), pp. 279–337

work page 2012

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H. S. Wilf.Generatingfunctionology.Boston, MA: Academic Press, 1994

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Eine arithmetische Eigenschaft automorpher Formen zu gewissen nicht- arithmetischen Gruppen

J. Wolfart. “Eine arithmetische Eigenschaft automorpher Formen zu gewissen nicht- arithmetischen Gruppen”. German. In:Math. Ann. 262 (1983), pp. 1–21

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Modularity of Calabi-Yau varieties: 2011 and beyond

N. Yui. “Modularity of Calabi-Yau varieties: 2011 and beyond”. In: Arithmetic and geometry ofK3surfaces and Calabi-Yau threefolds. Proceedings of the workshop, Toronto, Canada, August 16–25, 2011. Springer, New York, NY, 2013, pp. 101–139

work page 2011

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On quasi-modular forms, almost holomorphic modular forms, and the vector-valued modular forms of Shimura

S. Zemel. “On quasi-modular forms, almost holomorphic modular forms, and the vector-valued modular forms of Shimura”. In:Ramanujan J. 37.1 (2015), pp. 165–180. 26

work page 2015