Let (\omega=\omega_2\in\widehat{\mathbf Z}) be the idempotent whose (\mathbf Z_2)-component is (1) and whose (\mathbf Z_\ell)-component is (0) for (\ell\ne2). Use [ g^h=h^{-1}gh,\qquad [g,h]=g^{-1}h^{-1}gh. ] Define [ \sigma_2=\sigma^\omega,\qquad a=(x_0^{-3}\tau)^\omega,\qquad c=[x_1,x_1^{\sigma_2}] ] and let (\Gamma_R) be the profinite group defined by [ \tau^\sigma=\tau^2,\qquad r_R:=(x_0^\sigma)^{-1}a,x_1^2c=1, ] with the additional requirement that the closed normal closure of (x_0,x_1) be pro-(2).
The claim of this note is:
Theorem (relative to the results proved in the Roe–Turturean paper). Replacing the paper’s candidate relator by (r_R), all source-side hypotheses of its finite-quotient induction still hold. Consequently [ \Gamma_R\cong G_{\mathbf Q_2}. ]
Thus the work needed for a complete adaptation is confined to the candidate-specific calculations in Sections 3, 5, and 6 of the paper. The boundary-framed induction, finite Fourier inversion, exact-image recursion, and profinite reconstruction then apply unchanged.
References:
- Roe’s candidate presentation
- The Roe–Turturean paper
- Strategy of the proof
- Complete source interface
Kill the wild normal subgroup generated by (x_0,x_1). In every finite quotient, the relation (\tau^\sigma=\tau^2) forces (\tau) to have odd order: conjugation preserves its order, while squaring preserves that order only when it is odd. Hence (\tau^\omega=1). The second relation becomes trivial, and the tame quotient is [ \mathcal T=\langle \sigma,\tau\mid \tau^\sigma=\tau^2\rangle. ]
The assignment [ \nu_R(\sigma)=1,\qquad \nu_R(\tau)=\nu_R(x_0)=\nu_R(x_1)=0 ] defines a continuous epimorphism (\Gamma_R\twoheadrightarrow\mathbf Z_2). It induces the usual unramified marking on the tame quotient.
This supplies the tame half of the paper’s boundary construction.
In a pro-(2) quotient the tame relation forces (\tau=1), since
squaring cannot be an automorphism of a nontrivial procyclic pro-(2)
group. Also (g^\omega=g) for every pro-(2) element (g). Put
[
s=\sigma,\qquad x=x_0,\qquad y=x_1.
]
The maximal pro-(2) quotient is therefore
[
D_R=
\left\langle s,x,y\ \middle|
r_2=(x^s)^{-1}x^{-3}y^2[y,y^s]=1
\right\rangle_{\mathrm{pro}-2}.
]
Since (x^s=x[x,s]),
[
r_2=[x,s]^{-1}x^{-4}y^2[y,y^s].
]
Modulo the third Zassenhaus term, (x^{-4}) has degree at least (4) and ([y,y^s]) has degree at least (3). Thus the degree-two initial form is [ y^2+[x,s]. ] The corresponding cup–Bockstein matrix in the dual basis to (s,x,y) is [ \begin{pmatrix} 0&1&0\ 1&0&0\ 0&0&1 \end{pmatrix}, ] which is nonsingular. Hence (D_R) is a rank-three Demuškin group.
Its abelianization is [ B_R=D_R^{\mathrm{ab}} =\langle \bar s,\bar x,\bar y\mid -4\bar x+2\bar y=0\rangle_{\mathbf Z_2} \cong \mathbf Z_2^2\oplus C_2. ] Put [ t=\bar y-2\bar x. ] Then [ B_R=C_2t\oplus\mathbf Z_2\bar s\oplus\mathbf Z_2\bar x, \qquad \nu_R(t,\bar s,\bar x)=(0,1,0). ]
This is the main new calculation. Let (\chi_R) be the canonical Demuškin orientation and write [ S=\chi_R(s),\qquad X=\chi_R(x),\qquad Y=\chi_R(y). ] Labute’s characterization says that every crossed derivation of the free pro-(2) group into the rank-one module defined by (\chi_R) must descend through (r_2). If [ D(gh)=Dg+\chi_R(g)Dh, ] the coefficients of (D(r_2)) give [ \begin{aligned} Y^2&=X^4,\ 0&=-(X^{-1}S^{-1}+X^{-2}+X^{-3}+X^{-4}),\ 0&=S^{-1}\left[-X^{-1}(X-1)+\frac{(Y-1)^2}{Y^2}\right],\ 0&=X^{-4}(1+Y)+Y^{-1}(1-Y^{-1})(S^{-1}-1). \end{aligned} ] The branch (Y=X^2) gives no solution. The other branch gives [ Y=-X^2,\qquad X^3+2X^2+1=0,\qquad S=-\frac{X^3}{X^2+X+1}. ]
The polynomial (f(Z)=Z^3+2Z^2+1) has exactly one root in (\mathbf Z_2^\times): modulo (2) the only possible root is (1), and (f'(1)\equiv1\pmod2), so Hensel’s lemma applies. More precisely, [ X\equiv5\pmod {16},\qquad S\equiv13\pmod {16}. ] Therefore [ v_2(X-1)=v_2(S-1)=2, ] so each of (X,S) topologically generates (1+4\mathbf Z_2). Also [ \chi_R(t)=YX^{-2}=-1. ] It follows that [ \operatorname{im}\chi_R={\pm1}\times(1+4\mathbf Z_2). ] Labute’s classification now identifies (D_R) with the rank-three, (q=2), secondary-exponent-(4) group [ D_0=\langle A,S_0,Y_0\mid A^2S_0^4[S_0,Y_0]=1\rangle_{\mathrm{pro}-2}, ] which is (G_{\mathbf Q_2}(2)).
An unmarked isomorphism is not enough for the paper’s fibre product. We now construct the required isomorphism on abelianizations.
Because (X) and (S) both generate (1+4\mathbf Z_2), there is a unique (b\in\mathbf Z_2^\times) such that [ S=X^b. ] Set [ k=\bar s-b\bar x. ] Then [ B_R=C_2t\oplus\mathbf Z_2k\oplus\mathbf Z_2\bar x, \quad \nu_R(t,k,\bar x)=(0,1,0), \quad \chi_R(t,k,\bar x)=(-1,1,X). ]
For (D_0), put (t_0=\bar A+2\bar S_0) and (\eta=(-3)^{-1}). The paper proves [ D_0^{\mathrm{ab}} =C_2t_0\oplus\mathbf Z_2\bar S_0\oplus\mathbf Z_2\bar Y_0, ] with [ \nu_0(t_0,\bar S_0,\bar Y_0)=(0,1,0), \qquad \chi_0(t_0,\bar S_0,\bar Y_0)=(-1,1,\eta). ] Both (\eta) and (X) generate (1+4\mathbf Z_2), so there is a unit (u\in\mathbf Z_2^\times) with (\eta^u=X). Hence [ t\longmapsto t_0,\qquad k\longmapsto\bar S_0,\qquad \bar x\longmapsto u\bar Y_0 ] is an isomorphism (B_R\to D_0^{\mathrm{ab}}) preserving both (\nu) and (\chi).
Choose any abstract isomorphism (D_R\to D_0). Its abelianized map and the displayed desired map both preserve the intrinsic orientation. Their difference is therefore an orientation-preserving automorphism of (D_0^{\mathrm{ab}}). By the paper’s Proposition 3.9, that automorphism lifts to an automorphism of (D_0). Correcting the abstract isomorphism by this lift gives [ (D_R,\nu_R)\cong(G_{\mathbf Q_2}(2),\nu_{\mathrm{ur}}) ] as fully marked pairs over the entire (\mathbf Z_2), not merely modulo (2).
See:
- Standard dyadic group and orientation
- Lifting orientation-preserving abelian automorphisms
- Fully marked tame and pro-(2) quotients
Let (V) be a simple characteristic-(2) module for a finite lower tame quotient. Write the actions of (\sigma,\tau,\sigma_2) as [ \mathsf S,\qquad \mathsf T,\qquad \mathsf U=\mathsf S^\omega. ] Let [ P=1+\mathsf T+\cdots+\mathsf T^{e-1} ] be the tame-inertia norm projector, and write lift variables on ((\sigma,\tau,x_0,x_1)) as ((a,b,c,d)).
The tame row is the paper’s unchanged row [ L_{\mathrm t} =\mathsf S^{-1}(1+\mathsf T)a+ (\mathsf S^{-1}+1+\mathsf T)b. ] For Roe’s wild word:
- ((x_0^\sigma)^{-1}) contributes (\mathsf S^{-1}c);
- ((x_0^{-3}\tau)^\omega) contributes (P(c+b)), since (-3\equiv1\pmod2);
- (x_1^2) and ([x_1,x_1^{\sigma_2}]) have zero first derivative.
Thus [ L_{\mathrm w}=Pb+(P+\mathsf S^{-1})c ] and [ d^1_R= \begin{pmatrix} \mathsf S^{-1}(1+\mathsf T)& \mathsf S^{-1}+1+\mathsf T&0&0\ 0&P&P+\mathsf S^{-1}&0 \end{pmatrix}. ] This is exactly equation (5.4) of the paper after swapping its two wild columns (c,d).
For a nontrivial simple module:
- if (\mathsf T=1), then (P=1), the rows force (b=c=0), and a coboundary kills (a);
- if (V^{\mathsf T}=0), then (P=0), the rows force (c=0), a coboundary kills (b), and the tame row then forces (a=0).
In both cases every (H^1)-class has a unique representative [ (a,b,c,d)=(0,0,0,d). ]
For the trivial module, both relation rows have mod-(2) exponent vector equal to the (\tau)-coordinate, so [ d^1(a,b,c,d)=(b,b). ] The scalar cup–Bockstein form on (H^1), in coordinates (a,c,d), is [ a,c'+c,a'+d,d', ] which is the local Hilbert form after a change of basis.
See:
On the normalized representatives supported on (x_1), the only second-order contributions are:
- (x_1^2), giving the ordinary square diagonal;
- ([x_1,x_1^{\sigma_2}]), giving the two mixed terms involving (\mathsf U) and (\mathsf U^{-1}).
Consequently the degree-one pairing with (\lambda\in V^\vee) is [ (d,\lambda)\longmapsto \begin{cases} \lambda(d),&\mathsf T=1,\[1mm] \lambda!\left((1+\mathsf U+\mathsf U^{-1})d\right), &V^{\mathsf T}=0. \end{cases} ] In the ramified case (\mathsf U) has (2)-power order, so its minimal polynomial divides a power of (Z+1). Since (Z^2+Z+1) is coprime to (Z+1), [ 1+\mathsf U+\mathsf U^{-1} =\mathsf U^{-1}(\mathsf U^2+\mathsf U+1) ] is invertible. This is exactly the pairing operator in the paper.
The finite-word Stokes argument also applies. Modulo (2), the exponent vectors of the tame and wild relators are both [ (0,1,0,0). ] Their sum is zero, which is precisely the endpoint condition needed for the degree-(2) trace ((u_{\mathrm t},u_{\mathrm w})\mapsto u_{\mathrm t}+u_{\mathrm w}). The paper’s natural chain map is therefore a quasi-isomorphism for simple modules. Exact coefficient functoriality and cone dévissage extend it to every elementary, possibly nonsplit, module.
See:
Let (V) be nontrivial, simple, self-dual, and let (q:V\to\mathbf F_2) be an invariant nonsingular quadratic form with polar form (b_q). Evaluating Roe’s word on the normalized representative supported on (x_1) gives [ Q_R^0(d)=q(d)+b_q(d,\mathsf U^{-1}d). ] Equivalently, [ Q_R^0(d)= \begin{cases} q(d),&\mathsf T=1,\ q(d)+b_q(d,\mathsf U^{-1}d),&V^{\mathsf T}=0. \end{cases} ] Indeed, in the unramified case (\mathsf U=1) and (b_q(d,d)=0). In the ramified case its polar form is [ b_R(d,d') =b_q!\left(d,(1+\mathsf U+\mathsf U^{-1})d'\right). ]
These are exactly equations (6.25) and (6.26) of the paper, with the normalized wild coordinate renamed. Therefore the paper’s Hermitian-line and Wall-doubling arguments give [ #(Q_R^0)^{-1}(0)= \begin{cases} 2^{n-1}-2^{n/2-1},&V\text{ unramified},\ 2^{n-1}+2^{n/2-1},&V\text{ ramified}, \end{cases} \qquad n=\dim_{\mathbf F_2}V. ] These are the same zero counts, hence the same Gauss signs, as the local Galois obstruction.
See:
Sections 2–6 establish for (\Gamma_R) all four inputs of the paper’s complete source interface:
- the same fully marked tame and maximal pro-(2) boundary;
- the same lifting-space dimensions and lift multiplicities;
- the same natural perfect pairings and scalar cup–Bockstein form;
- the same quadratic Gauss sign on every self-dual simple head.
The later argument uses the source only through those inputs. Its finite Fourier transforms, exact-image recursion, central-cover corrections, and induction on the normal (2)-subgroup of a finite target therefore give [ |\operatorname{Sur}(\Gamma_R,H)| =|\operatorname{Sur}(G_{\mathbf Q_2},H)| ] for every finite group (H). The one-sided profinite reconstruction lemma then yields [ \boxed{\Gamma_R\cong G_{\mathbf Q_2}}. ]
See:
The argument above is substantially stronger than a comparison of low finite quotients: it identifies the exact source interface used by the paper’s all-finite-target induction. Still, a polished proof should record the following candidate-specific lemmas formally:
- the crossed-derivation computation of (\chi_R), including the exclusion of the branch (Y=X^2);
- the finite-word Fox ledger giving (L_{\mathrm w}=Pb+(P+\mathsf S^{-1})c);
- the Heisenberg expansion of (x_1^2[x_1,x_1^{\sigma_2}]);
- the trivial-module Stokes endpoint calculation.
Each is a short explicit identity. No new target-induction or local Galois-cohomology theorem appears to be needed.