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The arithmetic basilica: a quadratic PCF arboreal Galois group

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Pith's one-line read For f(z)=z^2−1, the arboreal Galois group G∞ always lies in an explicitly defined arithmetic basilica group M∞, and it equals M∞ exactly when a one-shot degree-16 field condition holds.

desk verdict A genuinely new computation of an arboreal Galois group for z^2-1, with a clean Frattini criterion; the only real soft spot is the compressed use of Pink's theorems in Theorem 5.1. read the letter →

arxiv 1909.00039 v4 pith:6TY2MT6W submitted 2019-08-30 math.NT math.DS

Faseeh Ahmad , Robert L. Benedetto , Jennifer Cain , Gregory Carroll , Lily Fang This is my paper · ORCID
classification math.NTmath.DS MSC 37P0511R3214G25
keywords arborealGaloisgrouppostcriticallyfinitepolynomialbasilicaarithmetic2-adiccyclotomiccharacteriteratedmonodromypreimagetree
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 establishes a complete Galois description for the iterated preimage tree of the quadratic map f(z)=$z^{2}$−1. It defines a group M∞, the arithmetic basilica group, which acts on the infinite binary tree and is an extension of the basilica group by Z_2^×. The main theorem shows that over any field K of characteristic different from 2, with root x0 not 0 or −1, the arboreal Galois group G∞ embeds into M∞, and equality holds exactly when a depth-five check or an equivalent degree-16 field condition holds. Because that condition holds for infinitely many rational x0, the full arithmetic basilica group is genuinely realized over Q. This matters because it gives a postcritically finite map whose arboreal Galois group is understood at every level, with the obstruction to fullness reduced to a single easy field extension.

What carries the argument

The central object is a function P(σ,x), taking values in the 2-adic units Z_2^×, attached to an automorphism σ of the binary tree and a node x. It is defined as (−1) to the parity of σ at x plus a weighted alternating sum of parities at selected higher nodes, and it computes the exact power to which σ sends primitive 2-power roots of unity: Theorem 2.2 proves σ(ζ)=$ζ^{{P(σ,y)}}$ for every Galois element and every node y. The arithmetic basilica group M∞ is then the subgroup of all tree automorphisms for which P(σ,x) is independent of x; the closed basilica group B∞, generated by the two recursively defined automorphisms α and β, is the kernel of P. The exact sequence and a Frattini-subgroup computation at level five carry the proof of the main equivalence.

What would settle it

Take K=Q and x0=1. Since [Q(√−1,√2,ζ8):Q]=4, the theorem predicts G∞ is not isomorphic to M∞ and in particular |G5|<$2^{25}$; computing the degree of the splitting field of $f^{5}$(z)−1 over Q would directly test this, and if that degree equals $2^{25}$, the main equivalence is false.

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Extended reading notes

Core claim

The central claim is the Main Theorem. Let K be a field of characteristic different from 2 and let x0∈K be nonzero and not −1. For f(z)=$z^{2}$−1, let G∞ be the Galois group of the union of all fields generated by iterated preimages of x0. Then G∞ is equivariantly isomorphic to a subgroup of M∞, and the following are equivalent: G∞≅M∞; the depth-five quotient G5 is isomorphic to M5; and the extension K(√−x0,√1+x0,ζ8) has degree 16 over K. The paper also shows M∞ sits in the exact sequence 0→B∞→M∞→Z_2^×→0 with B∞ the closed basilica group, and that for K=Q infinitely many x0 achieve the full group; the stated examples include x0 and −1−x0 in {5,6,10,11,12,13,14,19,...}.

Load-bearing premise

The argument depends on a cited function-field theorem that identifies the generic Galois group as an extension of the closed basilica group by the 2-adic units; the paper does not prove that theorem, and if it fails the main equivalence is unsupported.

Editorial extensions

If this is right

  • Whenever [K(√−x0,√1+x0,ζ8):K]=16, the full arithmetic basilica group occurs, and over K=Q this happens for infinitely many x0, including 5, 6, 10, 11, 12, 13, 14, 19, and their counterparts −1−x0.
  • Equality at depth five forces equality at every depth: G5≅M5 implies G∞≅M∞, so a finite computation certifies the entire infinite Galois group.
  • For every allowed K and x0, G∞ is confined to M∞, which is an extension of the closed basilica group by Z_2^×; the 2-adic cyclotomic character is the only arithmetic information beyond the basilica action.
  • The criterion is algorithmic: test the degree of the extension K(√−x0,√1+x0,ζ8), and if it is 16, the Galois group at level n has order 2^{m_n} with the explicit value from Theorem 4.6.

Reading between the lines

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

  • By the same mechanism, the depth-5 threshold is probably not special to z^2−1: the Frattini index 16 arises from the first two discriminants and ζ8, so for other PCF quadratics one should expect a finite depth, governed by the critical orbit and ramification in K∞, at which equality of finite-level groups forces equality at all levels.
  • The function P(σ,x) can be read as a 2-adic cyclotomic character computed from weighted parities; the same construction should produce an arithmetic iterated monodromy group for any PCF quadratic, giving a uniform family of model groups into which every specialization embeds.
  • Because the strong equality condition has degree 16, almost all rational x0 fail it; the paper's Conjecture 1 would then mean failure is only a finite-index defect. A natural quantitative check is to count x0 in growing boxes with [Q(√−x0,√1+x0,ζ8):Q]=16 and compare the density with the thin-set prediction.
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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

2 major / 3 minor

Summary. The paper defines a subgroup M∞ of the automorphism group of the infinite binary rooted tree, called the arithmetic basilica group, using a 2-adic quantity P(σ,x) built from the parities of a tree automorphism at certain nodes. For the postcritically finite quadratic polynomial f(z)=z^2−1, the authors prove that for any field K of characteristic not 2 and any root point x0∈K with x0≠0,−1, the arboreal Galois group G∞ embeds equivariantly into M∞. Their Main Theorem then asserts that G∞≅M∞ is equivalent both to the fifth-level condition G5≅M5 and to the explicit degree condition [K(√−x0,√1+x0,ζ8):K]=16. The proof strategy is to establish the isomorphism for the generic function field K=k(t) via two theorems of Pink, compute the Frattini subgroup of M∞ using the special case K=Q(t), and then use a Frattini argument to lift the isomorphism from a finite-level condition to the full pro-2 group. The paper also proves that for K=Q the degree condition holds for infinitely many x0 and formulates two broader conjectures about arboreal Galois groups of PCF maps.

Significance. If the main theorem is correct, it provides one of the first explicit, non-full arboreal Galois groups for a postcritically finite quadratic polynomial over number fields, together with a computationally checkable criterion for the full arithmetic basilica group to occur. The internal machinery in Sections 2–4 is a genuine contribution: the quantity P(σ,x) is defined combinatorially on Aut(T∞) without reference to Galois groups, the proof that M∞ is a subgroup and that P is a homomorphism is detailed, and the order computations for the finite quotients Mn, Bn, and En are explicit and agree with the table. The resulting condition (2c) is a clean, falsifiable prediction for infinitely many rational x0, and the paper appropriately credits Pink’s theorems as the external input connecting the combinatorial group to actual Galois groups. The main weakness is not the internal construction but the terseness and, in one place, an inclusion-direction error in the proof of the Frattini computation, which needs correction before the central equivalence can be regarded as fully proved.

major comments (2)
  1. [Section 5, Theorem 5.1, Eq. (28)] The proof of Theorem 5.1 states that restricting the injection G∞→M∞ yields an isomorphism Gal(K′∞/K′)≅B∞, and that the first square of the diagram commutes because this isomorphism is the restriction of the injection. As written, Gal(K′∞/K′) is not literally a subgroup of G∞: K′∞ = \bar{k}(t)(f^{-n}(t) for all n) is not a subfield of K∞, and the natural restriction map from Gal(\bar{k}K∞/\bar{k}K) to Gal(K∞/K) must be introduced to realize the geometric group as a subgroup of G∞. The paper should spell out this standard identification and verify explicitly that Pink’s Theorems 2.8.2 and 2.8.4 supply the compatibility of the left square of the diagram. Without that compatibility, the five-lemma conclusion that G∞→M∞ is an isomorphism does not follow from the two exact sequences.
  2. [Theorem 5.3, Step 2] The sentence "By the claim of Step 1, then, we have K^H_∞⊆L. Thus, the Frattini subgroup of G∞ is contained in Ψ := Gal(K∞/L)" has the inclusion reversed. From K^H⊆L it follows that H⊇Ψ for every maximal subgroup H, so the Frattini subgroup Φ contains Ψ rather than being contained in Ψ. The subsequent "Conversely" paragraph only proves Φ⊆Ψ, since Ψ is the intersection of the four displayed maximal subgroups. As printed, the proof of Theorem 5.3 establishes at most one inclusion and does not prove equality. The intended argument is clear, but the inclusion direction and the two inclusions needed for equality should be stated correctly.
minor comments (3)
  1. [Eq. (29) and Theorem 5.5] The stated value Δ2 = −64(1+x0)^2 x0 is off by a factor of 4; a direct discriminant computation for f2(z)−x0 = z4−2z2−x0 gives Δ2 = −256 x0(1+x0)^2. Since the square root of Δ2 is changed only by an element of K, the fields L and the parity arguments are unaffected, but the displayed formula should be corrected.
  2. [Theorem 5.5, Step 3] In the sentence describing β, the text says "again satisfies P(α)≡1 (mod 8)"; this should read P(β)≡1 (mod 8).
  3. [Theorem 5.1] Even after the identification of the geometric group with a subgroup of G∞ is made explicit, the map ρ in sequence (28) should be defined as the cyclotomic character on 2-power roots of unity, and the hypothesis [k(ζ8):k]=4 should be used to justify that the image of the cyclotomic character is all of Z2^× rather than merely a subgroup.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the combinatorial definition of M∞ and the cyclotomic-character theorem are self-contained; the equality results rest on external theorems of Pink, not on self-citation.

full rationale

The paper's central construction is not circular. M∞ is defined purely combinatorially in Definition 3.1 via the quantity P(σ,x) from Definition 2.1, which is built from parities of arbitrary tree automorphisms and involves no reference to Galois groups. Theorem 2.2 then proves, using the explicit labeling constructed in Lemma 1.4 and the elementary Lemma 1.2, that for a Galois element σ the cyclotomic action satisfies σ(ζ)=ζ^{P(σ,y)}; this is a genuine derivation, not a restatement of the definition. Consequently Corollary 3.3 — that the arboreal Galois group embeds into M∞ — follows from the paper's own internal arguments. The equality conditions in the Main Theorem and Theorem 5.5 depend on Pink's Theorems 2.8.2 and 2.8.4, which are external mathematical results with stated hypotheses (algebraically closed coefficient fields, [k(ζ8):k]=4) and are not authored by the present paper; invoking them is legitimate independent support rather than circularity. The subsequent Frattini-subgroup argument (Proposition 5.2, Theorem 5.3) is also self-contained once those external inputs are granted. The self-citations that do occur — [4] for the analogous cubic case and [5] for a standard discriminant formula — are not load-bearing for the main theorem; the discriminant formula is additionally attributed to [1], and the cubic analogue is used only for context and conjecture. The skeptic's concern about whether Gal(K'_∞/K') is literally a subgroup of G∞ in Theorem 5.1 is a potential mathematical gap in applying Pink's exact sequence, but it is not a circularity: no conclusion is being assumed through its own definition or through a self-citation. The paper does not rename a known result as a new prediction, and no fitted parameter is later relabeled as a prediction. Overall, the derivation chain is self-contained modulo genuinely external theorems.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper contains no fitted numbers or empirical parameters. Its central theorem rests on cited theorems of Pink about profinite iterated monodromy groups, on a standard discriminant formula for iterates, and on Hilbert's irreducibility theorem. No new physical or structural entity is postulated without proof; the arithmetic basilica group M∞ is explicitly defined and then proved to be the Galois group under the stated condition, so it does not play the role of an invented entity. The main proof work is internal once these inputs are granted.

assumptions (6)
  • domain assumption Pink's Theorem 2.8.2: for K' = k̄(t), the arboreal Galois group Gal(K'∞/K') is isomorphic to the closed basilica group B∞, equivariantly.
    Used in Theorem 5.1 to identify the kernel of P with B∞ and to prove G∞≅M∞ for K=Q(t); cited from [25], not proved in this paper.
  • domain assumption Pink's Theorem 2.8.4: for K=k(t) with k not algebraically closed, G∞ is an extension of B∞ by the 2-adic cyclotomic character, and when [k(ζ8):k]=4 the image is all of Z_2^×, giving the exact sequence 0→B∞→G∞→Z_2^×→0.
    This exact sequence is the foundation of Theorem 5.1 and of the Frattini subgroup computation in Theorem 5.3; the paper relies on the surjectivity of the cyclotomic character without proving it.
  • domain assumption Pink's Proposition 2.3.1: log2|Bn| = 2^{n-1} - Σ_{m=0}^{n-1} 2^{n-1-m} floor(m/2).
    Used in Theorem 4.5 to compute |Bn| and prove Bn = B'n; cited from [25].
  • standard math Iterated discriminant formula: Δ_n = (-4)^{2^{n-1}} Δ_{n-1}^2 (f^n(0)-t).
    Used in Theorem 5.3 Step 1 to show only (t) and (t+1) ramify in K'_n/K'; cited to [1, Proposition 3.2] and [5, equation (1)].
  • standard math Hilbert irreducibility theorem: for K=Q, the set of x0 with [Q(√-x0,√(1+x0),ζ8):Q] < 16 is thin.
    Used to conclude that the degree-16 condition holds for infinitely many x0∈Q; stated after the Main Theorem.
  • standard math Standard pro-finite group fact: maximal subgroups of a pro-2 group have index 2, and a subgroup meeting all cosets of the Frattini subgroup is the whole group (Proposition 5.2 proves the latter).
    Used in Theorem 5.3 and Theorem 5.5 Step 2 to upgrade the finite data to G∞≅M∞; Proposition 5.2 is proved in the paper.

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Pith. "Pith review of The arithmetic basilica: a quadratic PCF arboreal Galois group." pith.science (2026). https://pith.science/paper/6TY2MT6W

@misc{pith2026190900039,
  author       = {Pith},
  title        = {Pith review of: The arithmetic basilica: a quadratic PCF arboreal Galois group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6TY2MT6W}},
  note         = {Machine review of arXiv:1909.00039}
}
abstract

The arboreal Galois group of a polynomial $f$ over a field $K$ encodes the action of Galois on the iterated preimages of a root point $x_0\in K$, analogous to the action of Galois on the $\ell$-power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 - 1$ when the field $K$ and root point $x_0$ satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of $f$. For $K=\mathbb{Q}$, our condition holds for infinitely many choices of $x_0$.

Figures

Figures reproduced from arXiv: 1909.00039 by the authors.

Figure 1
Figure 1. Lemma 1.1: (α1α2) 2 = −y that we can choose k and x0 so that the inclusion of G∞ in G(φ, k) is an (equivariant) isomorphism. In the notation of Conjecture 2, one can also ask for sufficient conditions that G∞ has finite index in G(φ, k). Besides periodic x0, we also have [G(φ, k) : G∞] = ∞ if some Orb− φ (x0) contains a critical point of φ; if φ is not PCF, then this can happen for an infinite (but thin) set of x0 ∈… view at source ↗
Figure 2
Figure 2. Lemma 1.2 for m = 2. Proof. Write u = f(α1). Then α 2 1 = u + 1 and α 2 2 = −u + 1. See [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. A labeling of T3 Definition 1.3. A labeling of T∞ is a choice of two tree morphisms a, b : T∞ → T∞ such that a maps T∞ bijectively onto the subtree rooted at one of the two nodes connected to the root node x0, and b maps T∞ bijectively onto the subtree rooted at the other. For any integer n ≥ 1, a labeling of Tn is a choice of two injective tree morphisms a, b : Tn−1 → Tn with the same property. To see why the choic… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: P(σ, x) is a weighted sum of Par(σ, y) at the highlighted nodes y [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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