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Equivariant Modular Functions and Quantizations of Continued Fractions

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Pith's one-line read This paper proves exactly three representations of the modular group quantize the rationals, and at two golden-ratio parameters the new conjugate pair reproduces the branches of an anti-monotonic involution.

desk verdict Solid classification of PSL2(Z) q-representations; the Dyer/Jimm connection for the new pair is asserted, not proved, and under-specified. read the letter →

arxiv 2509.06036 v1 pith:WLQJ3G4Z submitted 2025-09-07 math.CO

Mustafa Topkara , A. Muhammed Uludag This is my paper · ORCID
classification math.CO MSC 11F0611A5505A30
keywords q-deformedrationalsequivariantmapsmodulargroupcontinuedfractionsouterautomorphismprojectivelinegoldenratiospecializationquantization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper classifies all quantizations of the modular group action on rationals by one-variable polynomials. With the translation $x\mapsto 1+x$ required to become $x\mapsto 1+qx$, there are exactly three matrix representations, one previously known and a new conjugate pair whose coefficients live in $\mathbb{Z}[\omega][q]$, $\omega=e^{2\pi i/6}$. For each representation there is a unique equivariant map from $\mathbb{P}^1(\mathbb{Z})$ to $\mathbb{P}^1(\mathbb{C}[q])$ once the value at $1$ is chosen between two explicit options, so the ambiguity in $q$-deforming a rational is a single binary choice. At the two golden-ratio parameters $q=-\varphi^2$ and $q=-\bar\varphi^2$, the known representation becomes conjugate to a distinguished outer automorphism of $\mathrm{PGL}_2(\mathbb{Z})$, and the equivariant maps become the two branches $J^\sharp$, $J^\flat$ of an anti-monotonic involution of the real line. The result matters because it shows the recently studied $q$-deformations of rationals are not isolated: they belong to a short list, and their degenerate specializations recover an order-reversing involution.

What carries the argument

The load-bearing object is the equivariant pair $(\Psi,\psi)$: a representation $\Psi:\mathrm{PSL}_2(\mathbb{Z})\to\mathrm{PGL}_2(\mathbb{C}(q))$ together with a function $\psi:\mathbb{P}^1(\mathbb{Z})\to\mathbb{P}^1(\mathbb{C}[q])$ intertwining the natural action through $\Psi$. The classification works by fixing $\Psi(T)=T=\begin{pmatrix}q&1\\0&1\end{pmatrix}$ and solving the two relations $S^2=1$ and $(TS)^3=1$ over $\mathbb{C}[q]$, which leaves exactly three matrix solutions. Existence and uniqueness of $\psi$ is decided by the stabilizer criterion of Lemma 3.1: because the action on $\mathbb{P}^1(\mathbb{Z})$ is transitive, $\psi$ is uniquely determined by $\psi(1)$, and it exists exactly when $\Psi(\mathrm{Stab}_{\mathrm{PSL}_2(\mathbb{Z})}(1))$ fixes $\psi(1)$; the stabilizer is generated by $A=TSTST^{-1}$, so the allowed values of $\psi(1)$ are the fixed points of $\Psi(A)$. The specialization theorem is then carried by the same machinery in reverse: a projective transformation $M$ is sought so that conjugation by $M$ sends the specialized generators $U_r,K_r,V_r$ to the specified outer automorphism, and $M\circ\psi_r$ automatically satisfies the functional equations of the involution branches.

What would settle it

Take the explicit projective transformations $M$ listed for $\Psi_+$ and $\Psi_-$ in Theorem 5.3, fix $q=-\varphi^2$, and compute $M\circ\psi_\pm(x)$ for a rational such as $x=2/3$; compare the result with $J^\sharp(x)$ (or $J^\flat(x)$) computed from the defining functional equations. A single mismatch at any rational would show the asserted conjugation to the involution is not correct, while systematic agreement would support it.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is a complete classification. Normalizing $\Psi(T)$ to be the map $x\mapsto 1+qx$, there are exactly three representations $\Psi:\mathrm{PSL}_2(\mathbb{Z})\to\mathrm{PGL}_2(\mathbb{C}(q))$: the known one with $\Psi(S)=\begin{pmatrix}0&-1\\q&0\end{pmatrix}$, and a conjugate pair with $\Psi_\pm(S)=\begin{pmatrix}1&q^{-1}\\-q+\omega^{\pm1}&-1\end{pmatrix}$, where $\omega=\exp(2\pi i/6)$. The representations extend to $\mathrm{PGL}_2(\mathbb{Z})$ (the conjugate ones degenerate at $q=\pm\omega$), and for each representation the equivariant map $\psi$ exists and is unique once $\psi(1)$ is chosen from the two fixed points of $\Psi(A)$ with $A=TSTST^{-1}$; the allowed values are $1$ and $q$ in the known case, and $\omega^{-1}$, $(1+\omega^2q)^{-1}$ (respectively $\omega$, $(1-\omega q)^{-1}$) in the two conjugate cases. Specializing to $q=-\varphi^2$ and $q=-\bar\varphi^2$ with $\varphi=(1+\sqrt5)/2$ and $\bar\varphi=-1/\varphi$, the paper exhibits projective transformations that conjugate the specialized representations to the outer automorphism of $\mathrm{PGL}_2(\mathbb{Z})$ given by $U\mapsto U$, $K\mapsto K$, $V\mapsto UV$, and it proves that the composed maps become exactly the two functions $J^\sharp$ and $J^\flat$ satisfying $J^\sharp(1)=\varphi^2$ and $J^\flat(1)=\bar\varphi^2$. For the two new conjugate representations the verification of this specialization is stated as similar to the known case.

Load-bearing premise

The paper leans on previously established properties of the real-line involution it calls $J_{\mathrm{imm}}$—continuous extension to all irrationals, matching one-sided limits, and the identity $J(J^\sharp(x))=x$—and for the two new conjugate representations the key conjugation is stated with a sketch rather than a full computation; if any of those facts fails, the advertised link to the outer automorphism collapses.

Editorial extensions

If this is right

  • The classification is complete for the chosen normalization: any representation satisfying $\Psi(T):x\mapsto 1+qx$ is one of the three listed, so the known quantization is not unique.
  • For each representation the equivariant map has exactly two possible normalizations, meaning the $q$-deformed rational $[x]_q$ is determined up to a binary choice of $\psi(1)$.
  • No equivariant function exists for the full group $\mathrm{PGL}_2(\mathbb{Z})$ with the natural normalization $\psi(1)=1$ or $q$; equivariance must be restricted to $\mathrm{PSL}_2(\mathbb{Z})$ or the target space must be enlarged.
  • At $q=-\varphi^2$ and $q=-\bar\varphi^2$, the specialized quantized groups are not contained in $\mathrm{PSL}_2(\mathbb{R})$, yet their index-two subgroups admit an exact sequence with projective determinant, giving explicit matrix generators for the kernels.
  • For transcendental specializations, $\Psi_r$ and $\Psi_r^\pm$ are isomorphisms onto their images, so the polynomial representations faithfully encode the modular group outside a countable set of algebraic exceptions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The conjugate pair likely gives rise to two new families of $q$-deformed rationals taking values in $\mathbb{Z}[\omega][q]$; these could be checked for the same palindromic trace symmetries that hold for the known representation.
  • The special values $q=-\varphi^2$ and $q=-\bar\varphi^2$ are the two real roots of $q^2+3q+1=0$; the appearance of the golden ratio suggests a boundary phenomenon where the quantized projective line degenerates onto an order-reversing involution, possibly tied to a maximally anti-monotonic map on the rationals.
  • Because the conjugation for $\Psi_\pm$ is only sketched, a direct computation would settle whether the new representations truly recover $J^\sharp$ and $J^\flat$ or instead produce nearby variants; if they do, the same mechanism may yield other rational-slope involutions at other algebraic specializations.
  • The paper's conjecture that $\mathrm{PSL}_2(\mathbb{Z},q=r)\simeq\mathrm{PSL}_2^\pm(\mathbb{Z},q=r)$ for every $r$ could be tested on the explicit root set of the palindromic polynomial in Example 2, where the two representations give identically collapsing elements.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper classifies group homomorphisms Psi : PSL2(Z) -> PGL2(C(q)) satisfying Psi(T) = [[q,1],[0,1]]. Theorem 2.1 establishes that there are exactly three such representations: the Morier-Genoud-Ovsienko representation and a conjugate pair Psi_± whose S-images are [[1,q^{-1}],[-q+omega^{±1},-1]]. The paper then constructs the associated PSL2(Z)-equivariant maps psi : P^1(Z) -> P^1(C[q]) using a stabilizer argument (Lemma 3.1, Propositions 3.2 and 3.3), studies specializations of q, and in Section 5 connects the representations at q = -phi^2 and -phi-bar^2 to Dyer's outer automorphism and the involution Jimm, with Theorem 5.3 asserting explicit conjugacies and identities M ∘ psi = J^sharp or J^flat.

Significance. If the results hold in full, the paper gives a clean and complete classification of the quantizations of PSL2(Z) with the fixed T-action, and it produces new conjugate quantizations of rationals. The main classification argument (Theorem 2.1) is a thorough case analysis that appears complete, and the existence/uniqueness proofs via Lemma 3.1 are rigorous. The explicit matrices and the numerical examples are useful. The advertised connection to Dyer's outer automorphism and Jimm is, however, only fully demonstrated for the Morier-Genoud-Ovsienko representation; for the two new conjugate representations the proof is deferred with no computation, and the statement does not specify which of the two equivariant map branches is used. This makes the central advertised application to Jimm unverifiable as written.

major comments (3)
  1. [Section 5, Theorem 5.3(2)-(3)] Parts (2) and (3) of Theorem 5.3 are asserted with the sentence 'The proof is similar to the first case' and no computation is shown. These identities M ∘ ψ±_r = J^sharp or J^flat are load-bearing for the paper's advertised connection to Dyer's automorphism and the involution Jimm. Unlike part (1), the reader is given no way to verify the conjugation of the generators or the functional equations. Please provide the full computation for both conjugate representations, or at least an explicit verification that the stated Möbius maps conjugate U_r, K_r, V_r to U, K, UV and that the resulting maps satisfy the defining equations of J^sharp and J^flat.
  2. [Section 5, Theorem 5.3(2)-(3) and Proposition 3.3] The statement of Theorem 5.3 is under-specified. Proposition 3.3 shows that for each of Ψ_+ and Ψ_- there are two distinct PSL2(Z)-equivariant maps, distinguished by the choice of ψ±(1): for Ψ_+ the choices are ω^{-1} and 1/(1+ω^2 q), and for Ψ_- the choices are ω and 1/(1-ω q). Theorem 5.3(2)-(3) writes M∘ψ+_{r}=J^sharp etc. without saying which branch is used for J^sharp and which for J^flat. Since the two branches are different functions, the identity can hold for at most one branch. Please specify the branch in each identity and verify, via Lemma 3.1 or directly, that the resulting map is the unique equivariant map for that branch.
  3. [Section 5, Theorem 5.3(2)-(3)] The 'if and only if' claim in parts (2) and (3) also requires an 'only if' argument showing that no other pair (r, M) can conjugate the representation to Dyer's outer automorphism. In part (1) this is obtained by solving the resulting equations; in parts (2) and (3) no such analysis is present. Without it the reader cannot verify the completeness of the listed pairs (r,M).
minor comments (4)
  1. [Section 3.1, proof of Proposition 3.2] The expression 'Stab_{PGL2(C)}(ψ(1))' should presumably be the stabilizer in the automorphism group of the target space, i.e. PGL2(C(q)), since ψ(1) is q-dependent; as written the notation is misleading.
  2. [Section 5.1] In the definition of Dyer's automorphism, the equality α(T)=TU uses that U and V commute; this is true because (UV)^2=1, but it would be helpful to state this explicitly.
  3. [Table 1] The entries for Ψ±(K) and Ψ±(U) are visually cluttered and the matrix for K± appears to have a misaligned row; please reformat for readability.
  4. [Section 4] In Proposition 4.1, the hypothesis 'not algebraic' is used as 'transcendental'; this is fine, but the wording 'not algebraic' is unusual and could be replaced by 'transcendental' for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three-representation classification and equivariant-map existence are derived from the PSL2(Z) presentation, with self-citations used only as context.

full rationale

The central derivation chain is self-contained. Section 2 fixes T(x)=1+qx from the quantization relation psi(1+m)=1+q psi(m) and then solves S^2=(TS)^3=1 by direct matrix computation, obtaining exactly the Morier-Genoud-Ovsienko representation and the conjugate pair; no parameter is fitted and no external theorem is assumed. Section 3 constructs the equivariant maps from the stabilizer criterion (Lemma 3.1), computing the fixed points of the stabilizer generator A (Propositions 3.2 and 3.3) rather than importing the target maps. Theorem 5.2 proves the existence and uniqueness of J^sharp and J^flat from the same stabilizer argument. Theorem 5.3(1) is derived by imposing the Jimm functional equations on a Mobius transform of psi and solving the resulting polynomial system; the special values r=-phi^2 and r=-bar-phi^2 are outputs of that computation, not inputs. The authors' self-citations [5]-[7] supply background and the real-line extension properties of Jimm, but those properties are not used in the classification or in the existence/uniqueness proofs, so they are not load-bearing for the central claim. Two verification gaps are flagged but are not circularity: Proposition 2.2 is dismissed as a 'routine check', and Theorem 5.3(2)-(3) defer the conjugation argument with 'the proof is similar to the first case'; these are omitted computations, not reductions of a prediction to its inputs, and the main classification is unaffected.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central classification is self-contained and uses only standard group presentations and field facts. The specialization-to-Jimm section imports two external results from the prior literature (the Jimm involution and the equivariant extension of the MGO map); these are not re-derived but are published results.

assumptions (4)
  • standard math Standard presentation of PGL2(Z) = <U,V,K> and PSL2(Z) = <S,T> with the relations used in Section 2.
    Cited [9]; used as the starting point for the representation classification.
  • standard math An involutive element in PGL2 over a field is represented by a trace-zero matrix.
    Used to derive the forms of V and V+/- in Section 2.
  • domain assumption The Jimm involution from [5] has the stated properties (existence on Q+, extension to R\Q, limits, and the relation J(J^sharp(x)) = x).
    Assumed from the authors' prior work [5] in Section 5; load-bearing for Theorem 5.3 but not for the Section 2-3 classification.
  • domain assumption Jouteur's result [4] that the MGO quantization map extends to R\Q and is PGL2(Z)-equivariant.
    Cited in Section 3.1; used to contrast with the failure on P^1(Z).

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Cite this review

Pith. "Pith review of Equivariant Modular Functions and Quantizations of Continued Fractions." pith.science (2026). https://pith.science/paper/WLQJ3G4Z

@misc{pith2026250906036,
  author       = {Pith},
  title        = {Pith review of: Equivariant Modular Functions and Quantizations of Continued Fractions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLQJ3G4Z}},
  note         = {Machine review of arXiv:2509.06036}
}
read the original abstract

We show that, in addition to the quantizations of the rational numbers discovered by Morier-Genoud and Ovsienko, there exist a pair of conjugate representations of the modular group and the corresponding equivariant maps with respect to these representations. We discuss some specializations of these quantizations and discuss its connection to Dyer's outer automorphism and the associated involution Jimm of the real line.

Figures

Figures reproduced from arXiv: 2509.06036 by the authors.

Figure 1
Figure 1. The locus where X := (T 2 q SqT 3 q SqT 5 q SqT 7 q Sq) 5 collapses to identity. Observe first the cyclotomic factor, which shows Ψr(X) = I inside PSL2(Z, q = exp 2mπi 5 ), m = 1, 2, 3, 4. Also observe that the main factor is a palindromic polynomial. Hence, its roots are symmetric with respect to the circle. There are no real roots in this case. For each root r, we have Ψr(X) = I. The existence of many roots on the… view at source ↗
Figure 2
Figure 2. Plot of the involution jimm on the unit interval. This involution J is induced by Dyer’s outer automorphism α of PGL2(Z) as we explain below. Dyer’s outer automorphism is also manifested as an automorphism of the Farey tree (the two-sided Stern-Brocot tree) of rationals [7], which ‘maximally violates’ the natural ordering of the nodes of the said tree, and also the natural ordering of its boundary. Hence, in a certa… view at source ↗
Figure 3
Figure 3. Plot of Ψr at r = exp 2πi 17  [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Plot of ψr(10) with x = 10 fixed while r traces the unit circle [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Plot of ψr([1, 1, 1, 1, 1, 1]) with x = [1, 1, 1, 1, 1, 1] fixed while r traces the unit circle [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

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Works this paper leans on

9 extracted references · 6 canonical work pages

  1. [1]

    Morier-Genoud and V

    S. Morier-Genoud and V. Ovsienko.q-deformed rationals and irrationals.arXiv preprint arXiv:2503.23834 (2025)

    arXiv 2025

  2. [2]

    Morier Genoud and V

    S. Morier Genoud and V. Ovsienko.On q-deformed real numbers.Experimental Mathematics 2022, Vol. 31, No. 2, 652–660

    work page 2022

  3. [3]

    Etingof,Onq-real andq-complex numbers.arXiv preprint arXiv:2508.08440 (2025)

    P. Etingof,Onq-real andq-complex numbers.arXiv preprint arXiv:2508.08440 (2025)

    arXiv 2025

  4. [4]

    Jouteur.Symmetries of the q-deformed real projective line.arXiv preprint arXiv:2503.02122 (2025)

    P. Jouteur.Symmetries of the q-deformed real projective line.arXiv preprint arXiv:2503.02122 (2025)

    arXiv 2025

  5. [5]

    A. M. Uluda˘ g and B. Eren G¨ okmen.The Conumerator and the Codenominator.Bulletin des Sciences Math´ ematiques, Volume 180, November 2022, 103192

    work page 2022

  6. [6]

    A. M. Uluda˘ g and H. Ayral.On the Involution Jimm.in: IRMA Lectures in Mathematics and Theoretical Physics, 2021, Vol. 33, pp. 561-578

    work page 2021

  7. [7]

    A. M. Uluda˘ g,On the involution Jimm.Topology and geometry–a collection of essays dedicated to Vladimir G. Turaev: 561-578

    work page

  8. [8]

    A. M. Uluda˘ g, A. Zeytin and M. Durmu¸ s.Binary quadratic forms as dessins.Journal de th´ eorie des nombres de Bordeaux 29.2 (2017): 445-469

    work page 2017

Show all 9 references
  1. [9]

    A. M. Uluda˘ g and A. Zeytin.A panaroma of the fundamental group of the modular orbifold.Handbook of Teichm¨ uller theory 6 (2016): 501-519. r= 2 r= 1/2 r= 3/2 r= 2/3 r= 4/3 r= 3/4 r= 5/4 r= 4/5 Table 2.Plots ofψ r for some positive real values ofr. r=−2 r=−1/2 r=−3 r= 1/3 r=−...

    2016

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