Unknown Book Proceedings Series Volume 00, XXXX

Harmonic and equianharmonic equations in the Grothendieck-Teichmüller group, II

Hiroaki Nakamura and Hiroshi Tsunogai

Abstract. In [LS], Lochak and Schneps introduced the “harmonic parameter g” of the Grothendieck-Teichmüller group ĜT. We closely study the behavior of g on the absolute Galois group G_{ℚ} using a family of lemniscate elliptic curves. We obtain a relationship of the adelic beta function and the harmonic parameter specialized in the matrix group SL_{2}(ℤ̂).

§1. Introduction.

Let ĜT be the profinite Grothendieck-Teichmüller group introduced and studied by Drinfeld [Dr], Ihara [I1]. It contains the Galois group G_{ℚ} = Gal(\overline{ℚ}/ℚ) and has the standard parametrization (λ, f) : ĜT ↪ ℤ̂^{×} × F̂_{2} (ℤ̂, F̂_{2} are respectively the profinite completions of the integer ring ℤ and of F_{2}, the free group generated by non-commutative symbols x and y.) In [NT], we closely studied Galois behaviors of the auxiliary parameters g and h : ĜT → F̂_{2} which had been introduced in Lochak-Schneps [LS] so as to decompose (uniquely) the main parameter f : ĜT → F̂_{2} (actually f terminates into the commutator subgroup [F̂_{2}, F̂_{2}] by definition) as follows:

f(x, y) = g(y, x)^{−1}g(x, y) = \{ \begin{matrix}y^{− \frac{λ−1}{2}} h(y, z)^{−1}h(x, y) & (λ ≡ 1 mod 6), \\ y^{− \frac{λ−1}{2}} h(y, z)^{−1}y^{−1}h(x, y) & (λ ≡ −1 mod 6), \end{matrix}

where z = (xy)^{−1} ∈ F̂_{2}. We showed in [NT] that the parameters g and h can be directly written by (λ, f) on the image of G_{ℚ} in ĜT, and presented several new-type equations satisfied by the Galois image.

In this paper, we continue our study with mainly concentrating on the parameter g. Our motivating problem to the present paper concerns with the matrix specialization of this parameter. In fact, in [N-I] Corollary 4.13, one of the authors explicitly computed the matrix f_{σ} ((_{0}^{1}_{1}^{2}), (_{−2}^{1} _{1}^{0})) ∈ SL_{2}(ℤ̂), which turned out later

\begin{matrix}c \\ ◯ \end{matrix}XXXX American Mathematical Society

1991 Mathematics Subject Classification. Primary 14H30; Secondary 11G05, 20E18, 20F36.

Key words and phrases. adelic beta functions, lemniscate elliptic curve.

in [NS] Remark 2.7 to be decomposed as the following “intriguing” form:

(∗) f_{σ} (( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} )) = (−1)^{\frac{λσ −1}{2}} ( \begin{matrix}1 & 0 \\ −8ρ_{2}(σ) & 1 \end{matrix} ) ( \begin{matrix}λ_{σ}^{−1} & 0 \\ 0 & λ_{σ} \end{matrix} ) ( \begin{matrix}1 & −8ρ_{2}(σ) \\ 0 & 1 \end{matrix} ) .

Here, ρ_{2} : G_{ℚ} → ℤ̂ designates the Kummer 1-cocycle along the positive roots of 2. This symmetric matrix should be decomposed into the form ^{t}G · G for some G corresponding to the image G = g_{σ} ((_{0}^{1}_{1}^{2}), (_{−2}^{1} _{1}^{0})). However, unlike the decomposition f(x, y) = g(y, x)^{−1}g(x, y) in F̂_{2}, the determination of G in SL_{2}(ℤ̂) does not follow from the decomposition property f_{σ}((_{0}^{1}_{1}^{2}), (_{−2}^{1} _{1}^{0})) = ^{t}G · G; namely, there remains apriori ambiguity modulo left multiplication by “rotation matrices” on G. In this paper, we shall determine the true choice of the matrix G representing g_{σ} ((_{0}^{1}_{1}^{2}), (_{−2}^{1} _{1}^{0})). Our answer is that the main factor is given by the special value of Anderson-Ihara’s adelic beta function B_{σ}(\frac{1}{4}, \frac{1}{4}). In terms of the measure version 𝔹_{σ} ∈ ℤ̂[[ℤ̂^{2}(1)]] of B_{σ} (see §2 below), we prove:

Theorem D. (§5 (5.5)) For each σ ∈ G_{ℚ}, we have

g_{σ} (( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} )) = (−1)^{\frac{λσ −1}{2}} 𝔹_{σ} (( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) · ( \begin{matrix}λ_{σ}^{−1} & −8ρ_{2}(σ)λ_{σ}^{−1} \\ 0 & (−1)^{\frac{λσ −1}{2}} \end{matrix} )

in GL_{2}(ℤ̂).

Once Theorem D is obtained, it is easy to recover (∗) by a basic property of 𝔹_{σ} (see (2.5.1)). The connection of the above two sides is, naively speaking, via the classical link between the harmonic ratio and the lemniscate elliptic curve Y^{2} = X^{3} − X. The right hand side reflects the adelic Galois version of the classical period integral via the beta function:

∫_{0}^{1} \frac{dt}{\sqrt{1 − t^{4}}} = \frac{1}{4\sqrt{2}}B(\frac{1}{4}, \frac{1}{4}).

Since the lemniscate elliptic curve is dominated by the Fermat quartic, it is not surprising that the Galois action on the Tate module is given by the adelic beta values at the pair of 4-th roots of unity. However, we wish to take an explicit choice of a basis of this Tate module which allows well controls to be linked with the Grothendieck-Teichmüller parameters. Thereby we construct in §3 an explicit G_{ℚ}- compatible immersion of the fundamental group of the (affine) lemniscate elliptic curve into the braid group with 4 strands. A close comparison of tangential basepoints on the braid configuration space makes clear the appearance of the harmonic ratio (0, \frac{1}{2}, 1, ∞) at an addressed object (§4). We settle the proof of Theorem D in §5.

In [NT] Theorem A, Proposition 6.1 (2), we proved the following equations in the braid group B̂_{3} = ⟨τ_{1}, τ_{2} | τ_{1}τ_{2}τ_{1} = τ_{2}τ_{1}τ_{2} =: η⟩ for the image of G_{ℚ} ↪ ĜT:

(GF_{0})  g(τ_{1}^{2}, τ_{2}^{2}) = η^{2ρ_{2}−ρ_{3}}f(τ_{1}, η)τ_{1}^{−2ρ_{2}+3ρ_{3}},
(GF_{1})  g(τ_{1}^{2}, τ_{2}^{2}) = f(τ_{1}^{2}, η)τ_{1}^{4ρ_{2}},
(GG)  g(τ_{1}^{2}, τ_{2}^{2}) = η^{2ρ_{2}−ρ_{3}}g(τ_{1}, τ_{2})τ_{1}^{−4ρ_{2}+3ρ_{3}}.

Here and henceforth, we sometimes omit σ from formulas for simplicity, when no confusion could occur from contexts. Combining Theorem D with the above (GF0), (GF1) and (GG) specialized with τ_{1} ↦ (_{0}^{1}_{1}^{1}), τ_{2} ↦ (_{−1}^{1}_{1}^{0}), we immediately obtain

HARMONIC EQUATIONS IN ĜT, II 3

Corollary E. On the image of G_{ℚ} in ĜT, we have

f (( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) = (−1)^{\frac{λ−1}{2}} 𝔹 (( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) · ( \begin{matrix}λ^{−1} & −12ρ_{2}λ^{−1} \\ 0 & (−1)^{\frac{λ−1}{2}} \end{matrix} ) .

f (( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) = (−1)^{\frac{λ−1}{2} −ρ_{2}}(_{−1}^{0} _{0}^{1})^{ρ_{3}}𝔹((_{−1}^{0} _{0}^{1}), (_{−1}^{0} _{0}^{1})) ( \begin{matrix}λ^{−1} & (−6ρ_{2}−3ρ_{3})λ^{−1} \\ 0 & (−1)^{\frac{λ−1}{2}} \end{matrix} ) ,

g (( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) = (−1)^{\frac{λ−1}{2} −ρ_{2}}(_{−1}^{0} _{0}^{1})^{ρ_{3}}𝔹((_{−1}^{0} _{0}^{1}), (_{−1}^{0} _{0}^{1})) ( \begin{matrix}λ^{−1} & (−4ρ_{2}−3ρ_{3})λ^{−1} \\ 0 & (−1)^{\frac{λ−1}{2}} \end{matrix} ) .

One obvious remaining subject is to pursue the analogy of these results for the equianharmonic ratio (0, e^{2πi/6}, 1, ∞). The lemniscate elliptic curve will then be replaced by the mordell elliptic curve Y^{2} = X^{4} − X (cf. (3.4)). We hope to continue the investigation in a subsequent work. In the present paper, all algebraic varieties are treated as defined over ℚ. We fix once and for all the embedding \overline{ℚ} ↪ ℂ. The notation ζ_{n} always represents the specified primitive n-th root of unity exp(2πi/n) ∈ ℂ.

§2. Short review of the adelic beta function.

(2.1) We shall start with the standard action of ĜT on the free profinite group F̂_{2} generated by the symbols x and y. For each σ ∈ ĜT associated are two parameters λ = λ_{σ} ∈ ℤ̂^{×} and f = f_{σ}(x, y) ∈ [F̂_{2}, F̂_{2}] which are unique to satisfy σ(x) = x^{λ}, σ(y) = f^{−1}y^{λ}f. The abelianization ℤ̂^{2}(1) of F̂_{2} is generated by the images x̄, ȳ of x, y respectively. The complete group algebra ℤ̂[[F̂_{2}]] and its abelianization ℤ̂[[ℤ̂^{2}(1)]] play fundamental roles in Ihara’s definition of the adelic beta function 𝔹_{σ}. Recall that in the former ring ℤ̂[[F̂_{2}]], one has free differential operators ∂_{x} := \frac{∂}{∂x}, ∂_{y} := \frac{∂}{∂y} as the profinite analog of Fox’s operators in the discrete case (cf. [I1] Appendix). Using them, one may associates the Jacobian matrix

J_{σ} := ( \begin{matrix}∂_{x}σ(x) & ∂_{y}σ(x) \\ ∂_{x}σ(y) & ∂_{y}σ(y) \end{matrix} ) (= ( \begin{matrix}\frac{x^{λ}−1}{x−1} & 0 \\ (∂_{x}f)(y^{λ} − 1) & \frac{y^{λ}−1}{y−1} + (∂_{y}f)(y^{λ} − 1) \end{matrix} ))

for each σ ∈ ĜT whose abelianization image J_{σ}^{ab} belongs to GL_{2}(ℤ̂[[ℤ̂^{2}(1)]]). Then,

Definition. (Ihara [I2])

𝔹_{σ}(x̄, ȳ) := \frac{x̄ − 1}{x̄^{λ} − 1} \frac{ȳ − 1}{ȳ^{λ} − 1} det(J_{σ}^{ab}) (= 1 + (∂_{y}f)^{ab}(ȳ − 1)).

Since ℤ̂[[ℤ̂^{2}(1)]] is identified with \underleftarrow{lim}_{n}(ℤ̂[x̄, ȳ]/(x̄^{n}−1, ȳ^{n}−1)), one may speak about the specialization 𝔹_{σ}(ζ, ζ′) where the variables x̄, ȳ are specialized to any roots of unity ζ, ζ′. The specialization is valued in ℤ̂ ⊗ ℤ(ζ, ζ′). As an element of ℤ̂[[ℤ̂^{2}(1)]], 𝔹_{σ} can be determined by the collection of all specialization values at pairs of roots of unity. G.Anderson [A] introduces the adelic beta function B_{σ} : (ℚ/ℤ)^{2} → ℤ̂ ⊗ ℚ^{ab} by setting B_{σ}(\frac{a}{n}, \frac{b}{n}) := 𝔹_{σ}(ζ_{n}^{a}, ζ_{n}^{b}) which is a closer analogue of the classical beta function B(\frac{a}{n}, \frac{b}{n}) := ∫_{0}^{1} t^{\frac{a}{n} −1}(1 − t)^{\frac{b}{n} −1}dt. These special values control the complex multiplication theory of the quotients of Fermat Jacobians, especially the Galois actions on the torsion points of them. In this paper, however, we will establish the control directly for the lemniscate elliptic curve in an elementary way, so that we do not enter details of Anderson’s deep device here. Also, we mainly use 𝔹_{σ} (the measure version of B_{σ}) for the usefulness of its Galois theoretic formations.

Some of the first basic properties of 𝔹_{σ} are:

𝔹_{σ}(x̄, ȳ) ∈ ℤ̂[[ℤ̂^{2}(1)]]^{×}, (B0) 𝔹_{σ}(x̄, ȳ) = 𝔹_{σ}(ȳ, x̄), (B1) 𝔹_{σ}(1, x̄) = 1, (B2)

𝔹_{σ}(x̄, x̄^{−1}) = \frac{1 − x̄}{1 − x̄^{λ}} · x̄^{\frac{λ−1}{2}} · λ_{σ}, (B3)

in particular, 𝔹_{σ}(−1, −1) = (−1)^{\frac{λ−1}{2}} λ_{σ}.

(2.2) An alternative useful characterization of 𝔹_{σ} may be given as follows. Let the commutator subgroup [F̂_{2}, F̂_{2}] be denoted F̂′_{2}, and let the double commutator subgroup [F̂′_{2}, F̂′_{2}] be denoted F̂′′_{2}. Then, the quotient F̂′_{2}/F̂′′_{2} is acted on by ℤ̂[[ℤ̂^{2}(1)]] via (linear extension of) conjugation, and Ihara showed that F̂′_{2}/F̂′′_{2} is a free cyclic ℤ̂[[ℤ̂^{2}(1)]]-module generated by [x, y] = xyx^{−1}y^{−1}. We shall write this conjugate action by ∗. Then, there exists a unique 𝔸_{σ}(x̄, ȳ) ∈ ℤ̂[[ℤ̂^{2}(1)]] such that

(FA)  f_{σ}(x, y) ≡ 𝔸_{σ}(x̄, ȳ) ∗ [x, y] mod F̂′′_{2}.

Then, the theory of Blanchfield-Lyndon sequence relates this quantity with the adelic beta function as follows:

(BA)  𝔹_{σ}(x̄, ȳ) = 1 + 𝔸_{σ}(x̄, ȳ) · (x̄ − 1)(ȳ − 1).

One remarkable use of 𝔸_{σ} is that if we define ρ_{N} : ĜT → ℤ̂ by

ρ_{N}(σ) := \frac{1}{N} ∑_{c=0}^{N−1}(ζ_{N}^{c} − 1)𝔸_{σ}(ζ_{N}^{c}, 1),

then, restricted on G_{ℚ} ⊂ ĜT, ρ_{N} gives the Kummer 1-cocycle on positive roots of N, i.e., σ(\sqrt[k]{N}) = ζ_{N}^{ρ_{N} (σ)} \sqrt[k]{N} for all k = 1, 2, ... and σ ∈ G_{ℚ}. From this, one can easily see that 𝔸_{σ}(−1, 1) = −ρ_{2}(σ) (see Appendix (A4); cf. also [NS] Proposition 5.3).

(2.3) Let 𝕎_{p} be the ring of restricted Witt vectors of 𝔽̄_{p}, and let 𝕎 = ∏_{p} 𝕎_{p} be the direct product for all primes p which is topologized by the collection {n𝕎}_{n∈ℕ} as the fundamental system of neighborhoods of 0. One can consider the enlarged ring 𝕎[[ℤ̂^{2}(1)]] naturally containing ℤ̂[[ℤ̂^{2}(1)]]. Ihara [I2] extends Anderson’s hyperadelic Gamma function IΓ_{σ} for all σ ∈ ĜT. In particular, he deduces that the 5-cyclic relation of ĜT allows the “Γ-decomposition” of 𝔹_{σ}:

(BΓ)  ∀σ ∈ ĜT, ∃ IΓ_{σ}(x̄) ∈ 𝕎[[ℤ̂]] s.t. 𝔹_{σ} = \frac{IΓ_{σ}(x̄) IΓ_{σ}(ȳ)}{IΓ_{σ}(x̄ȳ)}.

G.Anderson [A] proved that, for σ ∈ G_{ℚ}, the hyperadelic Gamma function IΓ_{σ} satisfies an analog of the Gauss n-multiplication formula:

(Γ_{n})  ∏_{nc=0} \frac{IΓ_{σ}(ζ_{n}^{c}x̄)}{IΓ_{σ}(ζ_{n}^{c})} · \frac{1}{IΓ_{σ}(x̄^{n})} = x̄^{nρ_{n}(σ)}.

It is unknown whether (Γ_{n}) holds on the total ĜT. Recently, Furusho [F] showed that the property (Γ_{2}) holds on a subgroup of ĜT satisfying certain geometric relations (IV) and (K_{2}). The following lemma indicates that the two main parameters λ and f of ĜT are “not totally independent.”

(2.4) Lemma. If σ ∈ ĜT satisifies (Γ_{2}) (e.g., σ ∈ G_{ℚ}), then,

(−1)^{ρ_{2}(σ)} = \frac{\sqrt{−1}^{\frac{λσ −1}{2}} (1 − \sqrt{−1}^{λ_{σ}})}{1 − \sqrt{−1}}.

Proof. For σ ∈ G_{ℚ}, one can separately check that each side coincides with 1 when λ_{σ} ≡ ±1 mod 8, and with −1 when λ_{σ} ≡ ±5 mod 8 (Notice that (−1)^{ρ_{2}(σ)} depends only on λ mod 8 since \sqrt{2} ∈ ℚ(ζ_{8}).) For general σ ∈ ĜT, the property (Γ_{2}) reads: \frac{IΓ_{σ}(x̄)IΓ_{σ}(−x̄)}{IΓ_{σ}(1)IΓ_{σ}(−1)} · \frac{1}{IΓ_{σ}(x̄^{2})} = x̄^{2ρ_{2}(σ)}. Putting x̄ = \sqrt{−1}, we then obtain

(−1)^{ρ_{2}(σ)} = (−1)^{−ρ_{2}(σ)} = \frac{𝔹_{σ}(−1, −1)}{𝔹_{σ}(\sqrt{−1}, −\sqrt{−1})}.

The lemma follows from this and (B3). □

(2.5) We finally complement a few observations on the specialization 𝔹_{σ}((_{−1}^{0} _{0}^{1}), (_{−1}^{0} _{0}^{1})) in the 2 by 2 matrix algebra M_{2}(ℤ̂). Let R be the commutative subring consisting of matrices of the form (_{b}^{a}_{a}^{−b}) (a, b ∈ ℤ̂). The issued 𝔹_{σ}((_{−1}^{0} _{0}^{1}), (_{−1}^{0} _{0}^{1})) is defined as the specialization image of 𝔹_{σ} under the map ℤ̂[[ℤ̂^{2}(1)]] → R (x, y ↦ (_{−1}^{0} _{0}^{1})). Then, for all σ ∈ ĜT,

(2.5.1)  ^{t}𝔹_{σ}((_{−1}^{0} _{0}^{1}), (_{−1}^{0} _{0}^{1})) · 𝔹_{σ}((_{−1}^{0} _{0}^{1}), (_{−1}^{0} _{0}^{1})) = (−1)^{\frac{λ−1}{2}} ( \begin{matrix}λ & 0 \\ 0 & λ \end{matrix} )

Proof. Since the matrix transposition gives an algebra automorphism on R, the result amounts to the formula 𝔹_{σ}(\sqrt{−1}^{−1}, \sqrt{−1}^{−1})𝔹_{σ}(\sqrt{−1}, \sqrt{−1}) = (−1)^{\frac{λ−1}{2}} λ. This is a simple result of (B2) (and (BΓ)). □

Since the transposed inverse of B = (_{b}^{a}_{a}^{−b}) (a, b ∈ ℤ̂) is det(B)^{−1}B, from (2.5.1) it is immediate to see

(2.5.2)  det 𝔹_{σ}((_{−1}^{0} _{0}^{1}), (_{−1}^{0} _{0}^{1})) = (−1)^{\frac{λ−1}{2}} λ.

§3. Lemniscate curve.

(3.1) In this section, we shall consider the lemniscate elliptic curve

E^{lem} : Y^{2} = X^{3} − X

as a cyclic branched cover over the projective t-line P_{t}^{1} ramified only over t = 0, 1, ∞. The fundamental group of E^{lem} − {O} is then recognized as a suitable subquotient of π_{1}(P_{t}^{1} − \{0, 1, ∞\}, \overrightarrow{01}). We recall that the standard action of σ = (λ, f) ∈ ĜT (where λ ∈ ℤ̂^{×}, f ∈ [F̂_{2}, F̂_{2}]) on the latter group is given as follows:

(3.1.1)  \{ \begin{matrix}x & ↦ x^{λ}, \\ y & ↦ f(y, x)y^{λ}f(x, y), \\ z & ↦ x^{\frac{λ−1}{2}} f(z, x)z^{λ}f(x, z)x^{\frac{1−λ}{2}} , \end{matrix}

where x, y, z are loops of π_{1}(P_{t}^{1} − \{0, 1, ∞\}, \overrightarrow{01}) with xyz = 1 chosen as in [NT] §1. The purpose of this section is to give explicit descriptions of the Galois representations in the Tate module of E^{lem} in terms of the adelic beta functions.

(3.2) The lemniscate elliptic curve E^{lem} : Y^{2} = X^{3} − X can be realized as a cyclic ramified cover of P_{t}^{1} of degree 4 by

X = \frac{1}{\sqrt{t}} and Y = \sqrt{\frac{1 − t}{\sqrt{t}^{3}}}.

The geometric fundamental group of E^{lem}−{O} is naturally regarded as a subgroup of π_{1}(P_{t}^{1} − {0, 1, ∞}; e_{1}|2, e_{∞}|4) = Δ̂(∞, 2, 4) with free generators x_{1} := x^{−1}z, x_{2} := xz^{−1}. The commutator [x_{1}, x_{2}] = x^{−4} generates an inertia subgroup over the point O ∈ E^{lem}. Therefore, if the extra condition ‘e_{0}|4’ is imposed on the above fundamental group of P_{t}^{1} − {0, 1, ∞}, then the finite-adelic Tate module T_{f}(E^{lem}) appears as the corresponding subgroup fitting in the commutative diagram of exact sequences:

1 ⟶ π_{1}(Elem ; \overline{ℚ} − \{O\}) ⟶ Δ̂(∞, 2, 4) ⟶ ℤ/4ℤ ⟶ 1
↓  ↓  ∥
1 ⟶ T_{f}(E^{lem}) ⟶ Δ̂(4, 2, 4) ⟶ ℤ/4ℤ ⟶ 1.

Note here that Δ(4, 2, 4) denotes the triangle group of Euclidean type corresponding to a plane tiling by isosceles right triangles which refines the regular square tessellation. Since the quotient π_{1}(P_{t}^{1} − {0, 1, ∞}; e_{0}|4, e_{1}|2, e_{∞}|4) is preserved by the operation of ĜT, the G_{ℚ}-action on the Tate module T_{f}(E^{lem}) is naturally extended to the ĜT-action on it.

(3.3) Proposition. Let x̄_{1}, x̄_{2} be the basis of the Tate module T_{f}(E^{lem}) which are the images of x_{1}, x_{2} respectively, and define the action of the ring ℤ̂[\sqrt{−1}] on T_{f}(E^{lem}) by \sqrt{−1} : x̄_{1} ↦ −x̄_{2}, x̄_{2} ↦ x̄_{1}. Then, each element σ = (λ, f) ∈ ĜT acts on T_{f}(E^{lem}) by

\{ \begin{matrix}x̄_{1} & ↦ (−1)^{\frac{λ−1}{2} +ρ_{2}(σ)}𝔹_{σ}(\sqrt{−1}, \sqrt{−1}) · x̄_{1}, \\ x̄_{2} & ↦ (−1)^{ρ_{2}(σ)}𝔹_{σ}(\sqrt{−1}, \sqrt{−1}) · x̄_{2}. \end{matrix}

Proof. First, we compute the action of σ = (λ, f) on x_{2} = xz^{−1}. The formula in (3.1) implies

σ(x_{2}) = x^{λ}x^{\frac{λ−1}{2}} f(z, x)z^{−λ}f(x, z)x^{\frac{1−λ}{2}}
= Int(x)^{\frac{λ−1}{2}} (Int(x)^{λ}(f(x, z)^{−1}) · x^{λ}z^{−λ}f(x, z)) .

We shall use the notation ≡ to designate the congruence modulo the kernel of the projection of Δ̂(∞, 2, 4) onto Δ̂(4, 2, 4). Since [x, z] ≡ x_{1}^{−1}x_{2} in T_{f}(E^{lem}), the formula (FA) of §2 implies that

f(x, z) ≡ (1 − \sqrt{−1})𝔸_{σ}(\sqrt{−1}, \sqrt{−1}) · x̄_{2}.

Here, note that the inner (conjugate) actions Int(x) and Int(z) on T_{f}(E^{lem}) are given by multiplication by \sqrt{−1}. We also have the congruence formula:

x^{λ}z^{−λ} ≡ \frac{1 − \sqrt{−1}^{λ}}{1 − \sqrt{−1}} · x̄_{2} (λ ∈ ℤ̂)

which can be easily proved by induction on λ ∈ ℤ_{>0} and by the standard continuity argument. Then, in total, it follows that

σ(x_{2}) ≡ \frac{\sqrt{−1}^{\frac{λ−1}{2}} (1 − \sqrt{−1}^{λ})}{1 − \sqrt{−1}} (1 + 𝔸_{σ}(\sqrt{−1}, \sqrt{−1})(1 − \sqrt{−1})^{2}) · x̄_{2}.

Since the value \sqrt{−1}^{\frac{λ−1}{2}} (1 − \sqrt{−1}^{λ})/(1 − \sqrt{−1}) is equal to (−1)^{ρ_{2}(σ)} (cf. Lemma (2.4)), we finally obtain

σ(x_{2}) ≡ (−1)^{ρ_{2}(σ)}𝔹_{σ}(\sqrt{−1}, \sqrt{−1}) · x̄_{2}.

To compute σ(x̄_{1}), we apply the above formula to x̄_{1} ≡ Int(x)(x_{2}). Then,

σ(x_{1}) ≡ Int(x^{λ})((−1)^{ρ_{2}(σ)}𝔹_{σ}(\sqrt{−1}, \sqrt{−1})·x̄_{2}) = \sqrt{−1}^{λ−1}(−1)^{ρ_{2}(σ)}𝔹_{σ}(\sqrt{−1}, \sqrt{−1})·x̄_{1}.

This completes the proof. □

(3.4) Before closing this section, we shall discuss briefly the case of the mordell elliptic curve E^{mor} : Y^{2} = X^{4} − X. This curve can be realized as a cyclic ramified cover of P_{t}^{1} of degree 6 by t = X^{3}: The function field of E^{mor} is generated by X = \sqrt[3]{t} and Y = \sqrt[6]{t}\sqrt{t − 1}. The geometric fundamental group of E^{mor} − \{O\} is naturally regarded as a subgroup of π_{1}(P_{t}^{1} − {0, 1, ∞}; e_{1}|2, e_{∞}|3) = Δ̂(∞, 2, 3) freely generated by x′_{1} := x^{−2}z, x′_{2} := x^{2}z^{−1}. An inertia group over the origin of E^{mor} is generated by the commutator [x′_{1}, x′_{2}] = x^{−6}. Therefore, if the extra condition ‘e_{0}|6’ is imposed on the fundamental group of P_{t}^{1} − {0, 1, ∞}, then the finite-adelic Tate module T_{f}(E^{mor}) appears as the corresponding subgroup. This fits in the following commutative diagram of exact sequences:

1 ⟶ π_{1}(Emor ; \overline{ℚ} − \{O\}) ⟶ Δ̂(∞, 2, 3) ⟶ ℤ/6ℤ ⟶ 1
↓  ↓  ∥
1 ⟶ T_{f}(E^{mor}) ⟶ Δ̂(6, 2, 3) ⟶ ℤ/6ℤ ⟶ 1.

The triangle group Δ(6, 2, 3) is of Euclidean type corresponding to a plane tiling by 30-60-90 triangles which refines the regular triangle tessellation. Since the quotient π_{1}(P_{t}^{1} − {0, 1, ∞}; e_{0}|6, e_{1}|2, e_{∞}|3) is preserved by the operation of ĜT, the G_{ℚ}- action on the Tate module T_{f}(E^{mor}) is naturally extended to the ĜT-action on it. By the similar argument to Proposition (3.3), we can show

(3.5) Proposition. Let x̄′_{1}, x̄′_{2} be the basis of the Tate module T_{f}(E^{mor}) which are the images of x′_{1}, x′_{2} respectively, and define the action of the ring ℤ̂[ζ_{6}] on T_{f}(E^{mor}) by ζ_{6} : x̄′_{1} ↦ (x̄′_{1} − x̄′_{2}), x̄′_{2} ↦ x̄′_{1}. Then, each element σ = (λ, f) ∈ ĜT acts on T_{f}(E^{mor}) by

\{ \begin{matrix}x̄′_{1} & ↦ ζ_{6}^{λ−1}𝔹_{σ}(ζ_{6}, ζ_{6}^{2}) · x̄′_{1}, \\ x̄′_{2} & ↦ 𝔹_{σ}(ζ_{6}, ζ_{6}^{2}) · x̄′_{2}.  □ \end{matrix}

We leave the proof as reader’s exercise. This result may be regarded as the Galois correspondent of the classical integral:

∫_{0}^{∞} \frac{dt}{\sqrt{t^{3} + 1}} = \frac{1}{3}B(\frac{1}{6}, \frac{1}{3}).

§4. Family of quartics and tangential base points.

(4.1) The purpose of this section is to give suitable embedding of P_{t}^{1} − {0, 1, ∞} to the space of quartics which enables us to consider the fundamental group of E^{lem} \ {O} in the framework of braid groups. Let A_{u}^{4} be the affine 4-space (over ℚ) where each point u = (u_{1}, u_{2}, u_{3}, u_{4}) is regarded as a quartic f_{u}(T) := T^{4} + u_{1}T^{3} + u_{2}T^{2} + u_{3}T + u_{4} of one formal variable T. Let D be the discriminant locus on A_{u}^{4} corresponding to the quartics with duplicate roots. Over the affine variety A_{u}^{4} − D, one has a Galois etale cover A_{v}^{4} − Δ called the space of solutions whose points are the ordered tuples v = (v_{1}, v_{2}, v_{3}, v_{4}) with distinct entries. (The divisor Δ denotes the hyper-diagonals of A_{v}^{4}.) The morphism of A_{v}^{4} − Δ to A_{u}^{4} − D is given by v ↦ f_{u}(T) := (T − v_{1})(T − v_{2})(T − v_{3})(T − v_{4}). Over A_{v}^{4} − Δ, one has a standard tangential basepoint b⃗ given by v = (0, t_{1}t_{2}t_{3}, t_{2}t_{3}, t_{3}) with values in the formal Puiseux power series ring \overline{ℚ}\{\{t_{1}, t_{2}, t_{3}\}\} (the maximal etale extension of \overline{ℚ}[[t_{1}, t_{2}, t_{3}]][(t_{1}t_{2}t_{3})^{−1}]). This is so called the Ihara-Matsumoto tangential basepoint ([IM]). Writing the image of b⃗ on A_{u}^{4} − D by the same symbol b⃗, one can naturally identify π_{1}(A_{u}^{4} − D, b⃗) = G_{ℚ} ⋉ B̂_{4}. The Galois action here can be extended to that of ĜT in the form of the famous formula of Drinfeld ([Dr], see [IM]), i.e., each σ = (λ, f) ∈ GT maps starndard generators as follows:

(4.1.1)  \{ \begin{matrix}τ_{1} & ↦ τ_{1}^{λ}, \\ τ_{2} & ↦ f(τ_{1}^{2}, τ_{2}^{2})^{−1}τ_{2}^{λ}f(τ_{1}^{2}, τ_{2}^{2}), \\ τ_{3} & ↦ f(η^{2}, τ_{3}^{2})^{−1}τ_{3}^{λ}f(η^{2}, τ_{3}^{2}). \end{matrix}

Here τ_{1}, τ_{2}, τ_{3} are standard generators of the braid group B_{4} (τ_{1}τ_{3} = τ_{3}τ_{1}, τ_{i}τ_{i+1}τ_{i} = τ_{i+1}τ_{i}τ_{i+1} (i = 1, 2)), and η = τ_{1}τ_{2}τ_{1} .

(4.2) The space A_{v}^{4}−Δ is an A^{1}-torsor by parallel transformations v ↦ (v_{1}+a, v_{2}+ a, v_{3} + a, v_{4} + a) (a ∈ A^{1}). This action does not affect the fundamental groupoid of A_{v}^{4} − Δ as π_{1}(A1 ; \overline{ℚ}) = 1. Two (tangential) basepoints b_{1}, b_{2} on A_{v}^{4} − Δ will be called equivalent (written b_{1} ∼ b_{2}) if they are transformed to each other by this A^{1}- action. The multiplicative group G_{m} acts on A_{v}^{4}−Δ by simultaneous multiplication on entries. The induced G_{m}-action on A_{u}^{4} − D is given by f_{u}(T) ↦ a^{4}f_{u}(a^{−1}T) (a ∈ G_{m}). We shall write those quotient spaces as

A_{v}^{4} := (A_{v}^{4} − Δ)/G_{m}, A_{u}^{4} := (A_{u}^{4} − Δ)/G_{m}.

Their geometric fundamental groups are given by moding out by the center ⟨ω_{4}⟩ (ω_{4} = (τ_{1}τ_{2}τ_{3})^{4}), i.e., π_{1}(A_{u}^{4} ⊗ \overline{ℚ}) is isomorphic to B̂_{4}^{∗} := B̂_{4}/⟨ω_{4}⟩, and π_{1}(A_{v}^{4} ⊗ \overline{ℚ}) is its pure part P̂_{4}^{∗} which is by definition the kernel of the “string-permutation” homomorphism B̂_{4}^{∗} → S_{4}. Remarkable observation here is that we can find braids in B̂_{4}^{∗} which behaves like the generators of the triangle group Δ(∞, 2, 4): One can take

(4.2.1)  η := τ_{1}τ_{2}τ_{1}, η_{2} := τ_{1}^{−1}τ_{2}^{−1}τ_{3}^{−1}τ_{1}^{−1}τ_{2}^{−1}τ_{1}^{−1}, η_{4} := τ_{1}τ_{2}τ_{3}

so that ηη_{2}η_{4} = 1, η_{2}^{2}(= ω_{4}^{−1}) = 1, η_{4}^{4}(= ω_{4}) = 1 in B̂_{4}^{∗}. Therefore, one has a homomorphism of Δ̂(∞, 2, 4) into B̂_{4}^{∗} with x ↦ η, y ↦ η_{2}, z ↦ η_{4}. Under this mapping, the generators x_{1}, x_{2} of the π_{1}(E^{∗} \ {O}) ⊂ Δ̂(∞, 2, 4) will be mapped into the braids ξ_{1}, ξ_{2} of B̂_{4}^{∗}:

(4.2.2)  \{ \begin{matrix}x_{1} = x^{−1}z & ↦ ξ_{1} := τ_{3}τ_{1}^{−1}, \\ x_{2} = xz^{−1} & ↦ ξ_{2} := τ_{2}τ_{1}τ_{3}^{−1}τ_{2}^{−1}τ_{3}τ_{1}^{−1} \end{matrix}

so that [x_{1}, x_{2}] ↦ [ξ_{1}, ξ_{2}] = ω_{4}η^{−4} = η^{−4}. It is known that these two braids generate a free profinite subgroup in B̂_{4}^{∗} (cf. §5 (5.2) below). From this one easily sees the injectivity of Δ̂(∞, 2, 4) ↪ B̂_{4}^{∗}.

(4.3) Now, let us introduce the embedding f^{lem} : P_{t}^{1} − {0, 1, ∞} → A^{4} \ D by giving the corresponding quartics as follows:

f_{t}^{lem}(T)(= f^{lem}(t)(T)) := T^{4} − 2T^{3} + \frac{3}{2} · \frac{t}{t − 1}T^{2} − \frac{1}{2} · \frac{t}{t − 1}T + \frac{1}{16} · \frac{t^{2}}{(t − 1)^{2}}

The discriminant of f_{t}^{lem}(T) is \frac{4t^{3}}{(t−1)^{6}} so that the morphism f^{lem} is well defined. The injectivity is obvious from the coefficients in T. The tasks we are now going to work on are:

(a) to find and fix a suitable path from b⃗ to the image of \overrightarrow{01}_{t} so that the image of loops in π_{1}(P_{t}^{1} − \{0, 1, ∞\}, \overrightarrow{01}_{t}) can be uniquely identified with loops in A_{u}^{4};

(b) to check the coincidence of the image of x, y with η, η_{2} respectively under

the above identification of (a). To approach (a), we shall solve the quartic equation f_{t}^{lem}(T) = 0 by the Cardano-Ferrari formula. The result is that the 4 zeros are given by

(4.3.1)  T = \frac{1}{2} + \sqrt{\frac{1}{4(1 − t)}} + \sqrt{\frac{1 + \sqrt{t}}{4(1 − t)}} + \sqrt{\frac{1 − \sqrt{t}}{4(1 − t)}}

where the (outer) three \sqrt{∗}’s are taken so that the product comes to be 1/8(1 − t). Near t being small > 0, after declaring that all \sqrt{∗} mean positive roots, one can specify 4 solutions as :

(4.3.2)  T_{1} ≈ −\frac{1}{2}\sqrt{t}, T_{2} ≈ 0, T_{3} ≈ \frac{1}{2}\sqrt{t}, T_{4} ≈ 2.

The corresponding lift of the tangential basepoint f^{lem}(\overrightarrow{01}_{t}) on A_{v}^{4} − Δ is then estimated as :

(0, t_{1}t_{2}t_{3}, t_{2}t_{3}, t_{3}) = (−\frac{1}{2}\sqrt{t}, 0, \frac{1}{2}\sqrt{t}, 2) ∼ (0, \frac{1}{2}\sqrt{t}, \sqrt{t}, 2 + \frac{1}{2}\sqrt{t}),

where ∼ means equivalence by A^{1}-action (cf.(4.2)); namely t_{1} = 1/2, t_{2} = \frac{\sqrt{t}}{2} =

\sqrt{\frac{t}{4}} and t_{3} ≓ 2. Moding out t_{3} ∈ G_{m}, we can define a path r on A_{v}^{4} from (the

image of) b⃗ to (that of) f^{lem}(\frac{1}{4} \overrightarrow{01}_{t}) by letting t_{1} move from \overrightarrow{01}_{t_{1}} to \frac{1}{2}. Let ε denote the straight path from \frac{1}{4} \overrightarrow{01}_{t} to \overrightarrow{01}_{t} on P_{t}^{1} − \{0, 1, ∞\}.

\frac{1}{4} \overrightarrow{01}_{t}  ⟶^{ε} \overrightarrow{01}_{t}(⇌ x, y)
↓_{f^{lem}}
\overrightarrow{b} ⟶^{r} f^{lem}(\frac{1}{4} \overrightarrow{01}_{t})

By using (the images on A_{u}^{4} of) these r and ε, the loops x, y of π_{1}(P_{t}^{1}−\{0, 1, ∞\}, \overrightarrow{01}_{t}) are mapped to the loops rf^{lem}(εxε^{−1})r^{−1}, rf^{lem}(εyε^{−1})r^{−1} of π_{1}(A_{u}^{4}, b⃗). For (b), it suffices to show that these correspond to the braids ξ_{1} and ξ_{2} of (4.2.2) respectively. The first one is not difficult: the image rf^{lem}(εxε^{−1})r^{−1} of x can be seen from (4.3.2) where t moves on a small circle around 0 in the anticlockwise way. The result is that T_{1} and T_{3} go around T_{2} to be interchanged to each other with leaving T_{2} (inside) and T_{4} (outside) invariant. This is nothing but the braid η = τ_{1}τ_{2}τ_{1}. When t is close to 1, the four roots T_{1},...,T_{4} behave approximately as follows:

T_{1} ≈ 1−\sqrt{\frac{1}{1 − t}}, T_{2} ≈ \frac{1 − \sqrt{2}}{2}\sqrt{\frac{1}{1 − t}}, T_{3} ≈ \frac{\sqrt{2} − 1}{2}\sqrt{\frac{1}{1 − t}}, T_{4} ≈ 1+\sqrt{\frac{1}{1 − t}}.

From this one sees that, when t goes on a small circle around 1 in the anticlockwise way, the points T_{1} and T_{4} (resp. T_{2} and T_{3}) move on a large circle clockwise with angle π to be switched to one another. This identifies rf^{lem}(εyε^{−1})r^{−1} with the braid η_{2} = τ_{1}^{−1}τ_{2}^{−1}τ_{3}^{−1}τ_{1}^{−1}τ_{2}^{−1}τ_{1}^{−1}.

We summarize the above discussion as

(4.4) Proposition. Let r and ε be paths introduced as above. Then, each σ ∈ G_{ℚ} acts on them by

\{ \begin{matrix}σ(r) & = g_{σ}(τ_{1}^{2}, τ_{2}^{2})^{−1} · r, \\ σ(ε) & = ε · x^{−2ρ_{2}(σ)}. \end{matrix}

The induced homomorphism (∗) ↦ rf^{lem}(ε(∗)ε^{−1})r^{−1} of π_{1}(P_{t}^{1} − \{0, 1, ∞\}, \overrightarrow{01}_{t}) into π_{1}(A_{u}^{4}) is injective, and maps x ↦ η, y ↦ η_{2} and z ↦ η_{4}. □

§5. Galois actions and proof of Theorem D.

(5.1) Proposition. Under the standard action of G_{ℚ}(⊂ ĜT) on B̂_{4} (4.1.1), the elements η := τ_{1}τ_{2}τ_{1}, η_{2} := τ_{1}^{−1}τ_{2}^{−1}τ_{3}^{−1}τ_{1}^{−1}τ_{2}^{−1}τ_{1}^{−1}, η_{4} := τ_{1}τ_{2}τ_{3}, ξ_{1} := τ_{3}τ_{1}^{−1} and ξ_{2} := τ_{2}τ_{1}τ_{3}^{−1}τ_{2}^{−1}τ_{3}τ_{1}^{−1} are explicitly transformed as follows:

(1)  η ↦ g(τ_{1}^{2}, τ_{2}^{2})^{−1}η^{λ}g(τ_{1}^{2}, τ_{2}^{2}), = f(τ_{2}^{2}, τ_{1}^{2})η^{λ}
(2)  η_{2} ↦ g(τ_{1}^{2}, τ_{2}^{2})^{−1}η^{−2ρ_{2}}f(η, η_{2})η_{2}^{λ}f(η_{2}, η)η^{2ρ_{2}}g(τ_{1}^{2}, τ_{2}^{2}),
(3)  η_{4} ↦ g(τ_{1}^{2}, τ_{2}^{2})^{−1}η^{\frac{λ−1}{2} −2ρ_{2}}f(η_{4}, η)η_{4}^{λ}f(η, η_{4})η^{− \frac{λ−1}{2} +2ρ_{2}}g(τ_{1}^{2}, τ_{2}^{2});
(4)  ξ_{1} ↦ η^{−8ρ_{2}}f(ξ_{1}, z_{0})ξ_{1}^{λ}f(z_{0}, ξ_{1})η^{8ρ_{2}},
(5)  ξ_{2} ↦ f(τ_{2}^{2}, τ_{1}^{2})η^{λ−1−8ρ_{2}}f(ξ_{2}^{−1}, z_{0})ξ_{2}^{λ}f(z_{0}, ξ_{2}^{−1})η^{1−λ+8ρ_{2}}f(τ_{1}^{2}, τ_{2}^{2}).

Here z_{0} = η^{4}ω_{4}^{−1} is the braid satisfying [ξ_{1}, ξ_{2}]z_{0} = 1.

Proof. The result (1) dates back to [N-I] Proposition 4.12. The validity of the first three formulae (1)∼(3) in B̂_{4}^{∗} are consequences of Proposition (4.4) and the G_{ℚ}-action on x, y, z described in (3.1). To lift them to B̂_{4}, note that the kernel of B̂_{4} → B̂_{4}^{∗} is the center ⟨ω_{4}⟩ of B̂_{4} and that the G_{ℚ}-actions on η^{4}, η_{2}^{2} = ω_{4}^{−1} and η_{4}^{4} = ω_{4} are given by their λ powers. The abelianization map B̂_{4} → ℤ̂ assures the coincidence of both sides of the issued lifts. (4) follows from the standard G_{ℚ}- action on ξ_{1} = τ_{3}τ_{1}^{−1} ∈ B̂_{3} according to Drinfeld’s formula (4.1.1) together with the relation (IV) first proved in [N-I] Theorem 4.16:

(IV) f(τ_{3}^{2}, η^{2}) = η^{−8ρ_{2}}f(ξ_{1}, η^{4}ω_{4}^{−1})τ_{1}^{4ρ_{2}}τ_{3}^{−4ρ_{2}} = η^{−8ρ_{2}}f(ξ_{1}, η^{4}ω_{4}^{−1})τ_{1}^{4ρ_{2}}ξ_{1}^{−4ρ_{2}}

Note that [ξ_{1}, ξ_{2}](η^{4}ω_{4}^{−1}) = 1. Applying (1),(4) to ξ_{2} = ηξ_{1}^{−1}η^{−1}, we obtain (5). □

(5.2) Let M_{1,2} be the moduli stack of smooth genus 1 curves with 2 ordered marked points and let f_{2} : M_{1,2} → M_{1,1} be the morphism to the moduli stack of elliptic curves which forgets the second marked points. Each quartic f_{u}(T) of A^{4} \ D gives the elliptic curve Y^{2} = f_{u}(T) with two ordered points at infinity. This gives rise to an isomorphism of fundamental groups π_{1}(A_{u}^{4}) →^{∼} π_{1}(M_{1,2}) (cf. [N-I] p.353, [N-II] §7.8). The homotopy exact sequence of this fibration is of the form:

π_{1}(A_{u}^{4})
↓^{isom.}
1 ⟶ F̂_{2} ⟶ π_{1}(M_{1,2}) ⟶^{f_{2}} π_{1}(M_{1,1}) ⟶ 1.

The kernel part F̂_{2} corresponds to the geometric fundamental group of a fibre elliptic curve minus origin. In [N-I] §4.9, we saw that ξ_{1}, ξ_{2} give a generator system of this F̂_{2}. Thus we see: ξ_{1}, ξ_{2} generates a free profinite subgroup in π_{1}(A_{u}^{4}), hence also in B̂_{4}.

(5.3) Remark. The above forgetful projection f_{2} can be lifted to π_{1}(A^{4} \ D_{4}) → π_{1}(A^{3} \ D_{3}) coming from a certain morphism F : A^{4} \ D_{4} → A^{3} \ D_{3}. This morphism F, called the Ferrari morphism (cf. [N01]) associates to quartic polynomials their resolvent cubics, and will play basic roles in the theory of Weierstrass tangential base points. (See the forthcoming paper partly based on [N01]).

(5.4) In [N-I], we studied various actions on the free subgroup F̂_{2} = ⟨ξ_{1}, ξ_{2}⟩ of π_{1}(M_{1,2}) ≅ B̂_{4}^{∗} which embodies the fundamental group of punctured elliptic curves. Geometric monodromy on them can be summarized as the inner actions by τ_{1}, τ_{2}. This action was determined in the following simple way:

(5.4.1)  Int(τ_{1}) : \{ \begin{matrix}ξ_{1} ↦ ξ_{1}, \\ ξ_{2} ↦ ξ_{2}ξ_{1}, \end{matrix} Int(τ_{2}) : \{ \begin{matrix}ξ_{1} ↦ ξ_{2}ξ_{1}^{−1}, \\ ξ_{2} ↦ ξ_{2}. \end{matrix}

Once given an elliptic curve with Weierstrass equation Y^{2} = 4X^{3} − g_{2}X − g_{3}, then we may associate in a standard way the Weierstrass tangential basepoint on it and the Galois action on F̂_{2} (cf. also [N98]). One important example is the case of the Tate elliptic curve

Tate(q) : Y^{2} = 4X^{3} − g_{2}(q)X − g_{3}(q)

over the formal Laurant power series field ℚ((q)), where g_{2}(q), g_{3}(q) are given as Eisenstein q-series of weight 4, 6 (constant terms are 1/12, −1/216) respectively. From the corresponding Weierstrass tangential basepoint, the Galois group G_{ℚ} acts by

(5.4.2)  \{ \begin{matrix}ξ_{1} ↦ f(ξ_{1}, z_{0})ξ_{1}^{λ}f(z_{0}, ξ_{1}), \\ ξ_{2} ↦ f(ξ_{1}, z_{0})x1^{\frac{1−λ}{2}} f(ξ_{2}ξ_{1}^{−1}ξ_{2}^{−1}, ξ_{1})ξ_{2}ξ1^{\frac{λ−1}{2}} f(z_{0}, ξ_{1}), \\ z_{0} ↦ z_{0}^{λ}, \end{matrix}

where [ξ_{1}, ξ_{2}]z_{0} = 1. The difference between this action and the Drinfeld-Ihara-Matsumoto action has been determined in [N-I] §4. (In fact, the conjugation by η^{−8ρ_{2}}τ_{1}^{8ρ_{2}} makes (5.4.2) coincide with Proposition (5.1) (4)-(5).) Using this circle of ideas, we showed in [N-I] Corollary 4.13:

(5.4.3) f_{σ} (( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} )) = (−1)^{\frac{λσ −1}{2}} ( \begin{matrix}1 & 0 \\ −8ρ_{2}(σ) & 1 \end{matrix} ) ( \begin{matrix}λ_{σ}^{−1} & 0 \\ 0 & λ_{σ} \end{matrix} ) ( \begin{matrix}1 & −8ρ_{2}(σ) \\ 0 & 1 \end{matrix} ) .

Now, we are ready to prove the following

(5.5) Theorem. For σ ∈ G_{ℚ}, we have the following equation:

g_{σ} (( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} )) = (−1)^{\frac{λσ −1}{2}} 𝔹_{σ} (( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) · ( \begin{matrix}λ_{σ}^{−1} & −8ρ_{2}(σ)λ_{σ}^{−1} \\ 0 & (−1)^{\frac{λσ −1}{2}} \end{matrix} ) .

Proof. Identify T_{f}(E^{lem}) with ℤ̂^{2} by x̄_{1} ↦ (_{0}^{1}), x̄_{2} ↦ (_{1}^{0}). By Proposition (3.3), the G_{ℚ}-action is given as the right multiplication by the matrix

(#)  (−1)^{ρ_{2}(σ)}𝔹_{σ} (( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) · ( \begin{matrix}(−1)^{\frac{λ−1}{2}} & 0 \\ 0 & 1 \end{matrix} ) .

On the other hand, by Proposition (5.1) (4)-(5), the standard action from b⃗ on the abelianization of F̂_{2} = ⟨ξ_{1}, ξ_{2}⟩ is given by

((−1)^{−4ρ_{2}} \binom{λ}{0}, f_{σ} (( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} ) , ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} )) (−1)^{\frac{λ−1}{2} −4ρ_{2}} \binom{0}{λ}) = ( \begin{matrix}λ & 8ρ_{2}(σ) \\ 0 & 1 \end{matrix} ) ,

Therefore, taking into consideration the factor coming from our path rf^{lem}(ε) from b⃗ to f^{lem}(\overrightarrow{01}_{t}), we see that the matrix (#) is equal to

(−1)^{ρ_{2}(σ)}g_{σ} (( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} )) ( \begin{matrix}λ & 8ρ_{2}(σ) \\ 0 & 1 \end{matrix} ) .

This completes the proof. □

Appendix. A note on the cocycle Ψ_{n}^{(0)}.

In [NT] Remark (6.3), we mentioned about Ihara’s 1-cocycle Ψ_{n}^{(0)} (n ∈ ℕ) whose minus sign −Ψ_{n}^{(0)} extends the Kummer 1-cocycle ρ_{n} on the positive roots of n. In fact, originated from a difference of path conventions between Ihara’s papers [I1,I2,I3] and ours, the association manner G_{ℚ} → ĜT delicately differs from each other so that the minus sign of “−Ψ_{n}^{(0)}” in that claim should be dropped if Ψ_{n}^{(0)} is replaced by its italicized version “Ψ_{n}^{(0)}” introduced in our convention exactly in the same way as Ψ_{n}^{(0)} in Ihara’s paper [I1]. To normalize ambiguities buried in contexts of literatures, in this appendix we shall give an explicit exposition starting from the definition of our italicized Ψ_{n}^{(0)} and then shall give a proof of the regularized statement:

(A1) Proposition. The 1-cocycle Ψ_{n}^{(0)} : ĜT → ℤ̂(1) extends the Kummer 1-cocycle ρ_{n} : G_{ℚ} → ℤ̂(1) attached to the real positive roots of n.

Let F̂_{2} be the free profinite group freely generated by the symbols x, y. For a positive integer n, let H_{n} be the kernel of the homomorphism F̂_{2} → μ_{n} defined by x ↦ ζ_{n}, y ↦ 1, which is a free profinite subgroup F̂_{n+1} freely generated by the x^{−a}yx^{a} (a ∈ ℤ/nℤ) and x^{n}. Since, for any σ = (λ_{σ}, f_{σ}) ∈ ĜT, f_{σ} belongs to F̂′_{2} ⊂ H_{n}, there exists a unique element (n + 1-term proword) f_{σ}^{(n)} = f_{σ}^{(n)}(ξ, η_{0}, η_{1}, …, η_{n−1}) ∈ F̂_{n+1} satisfying f_{σ}^{(n)}(x^{n}, y, xyx^{−1}, …, x^{n−1}yx^{−(n−1)}) = f_{σ}(x, y). For a ∈ ℤ/nℤ, consider the homomorphism ψ_{n}^{(a)} : F_{n+1} → ℤ̂ given by η_{a} ↦ 1 and ξ, η_{i} ↦ 0 (i ≠ a), and then:

(A2) Definition. Define Ψ_{n}^{(a)}(σ) := ψ_{n}^{(a)}(f_{σ}^{(n)}) (a ∈ ℤ/nℤ).

By the same argument as [I2] p.171, these Ψ_{n}^{(a)} can also be characterized by the formula on ĜT:

(A3) (\frac{𝔹_{σ}(x̄, ȳ) − 1}{ȳ − 1})_{ȳ=1} = (x̄ − 1)𝔸_{σ}(x̄, 1) ≡ ∑_{a∈ℤ/nℤ} Ψ_{n}^{(a)}(σ)x̄^{−a} mod (x̄^{n} − 1).

In particular,

(A4)  Ψ_{n}^{(0)}(σ) := \frac{1}{n} ∑_{c=0}^{n−1}(ζ_{n}^{c} − 1)𝔸_{σ}(ζ_{n}^{c}, 1).

To prove Proposition (A1), we shall compute Ψ_{n}^{(0)}(σ) for σ ∈ G_{ℚ}. Let X_{w} = P^{1} ∖ {0, μ_{n}, ∞} → X_{z} = P^{1} ∖ {0, 1, ∞} be the cyclic cover given by z = w^{n}, and consider the tangential base points \overrightarrow{01}_{z}, \overrightarrow{10}_{z} on the z-line and \overrightarrow{01}_{w}, \overrightarrow{10}_{w}, n\overrightarrow{10}_{w} on the w-line, whose value fields are \overline{ℚ}\{\{z\}\}, \overline{ℚ}\{\{z_{1}\}\} (z_{1} = 1 − z), \overline{ℚ}\{\{w\}\}, \overline{ℚ}\{\{w_{1}\}\} (w_{1} = 1 − w), \overline{ℚ}\{\{nw_{1}\}\} respectively. Let L\overrightarrow{01}_{z} (resp. L\overrightarrow{10}_{z}) be the maximal algebraic extention of ℚ(z) in \overline{ℚ}\{\{z\}\} (resp. \overline{ℚ}\{\{z_{1}\}\}) unramified outside 0, 1, ∞ (viz. over X_{z}), and let L\overrightarrow{01}_{w} (resp. L\overrightarrow{10}_{w}, L_{n\overrightarrow{10}_{w}}) be the maximal algebraic extention of ℚ(w) in \overline{ℚ}\{\{w\}\} (resp. \overline{ℚ}\{\{w_{1}\}\}, \overline{ℚ}\{\{nw_{1}\}\}) unramified outside 0, μ_{n}, ∞ (viz. over X_{w}). We shall connect the above tangential base points by the standard paths along the real axis:

(A5)  \{ \begin{matrix}p_{z} ∈ π_{1}(X_{z}; \overrightarrow{01}_{z}, \overrightarrow{10}_{z}) ≃ Isom_{ℚ(z)}(L\overrightarrow{10}_{z}, L\overrightarrow{01}_{z}), \\ p_{w} ∈ π_{1}(X_{w}; \overrightarrow{01}_{w}, \overrightarrow{10}_{w}) ≃ Isom_{ℚ(w)}(L\overrightarrow{10}_{w}, L\overrightarrow{01}_{w}), \\ δ_{n} ∈ π_{1}(X_{w}; n\overrightarrow{10}_{w}, \overrightarrow{10}_{w}) ≃ Isom_{ℚ(w)}(L\overrightarrow{10}_{w}, L_{n\overrightarrow{10}_{w}}). \end{matrix}

The covering map β_{n} : X_{w} → X_{z} and the path δ_{n} give an embedding π_{1}(X_{w}; \overrightarrow{01}_{w}) ↪ π_{1}(X_{z}; \overrightarrow{01}_{z}) such that ξ ↦ x^{n}, η_{i} ↦ x^{i}yx^{−i} (i = 0, 1, …, n − 1). Meanwhile, one has a natural open immersion ι_{n} : X_{w} → X_{z} (w ↦ z) which induces the surjection of π_{1}(X_{w}; \overrightarrow{01}_{w}) onto π_{1}(X_{z}; \overrightarrow{01}_{z}) such that ξ ↦ x, η_{0} ↦ y, η_{i} ↦ 1 (i = 1, …, n−1).

Writing the action of σ ∈ G_{ℚ} at a tangential base point b as σ_{b}, one may describe Galois effects on the above paths as follows:

(A6) \{ \begin{matrix}σ(p_{z}) & = σ\overrightarrow{01}_{z}p_{z}σ_{\overrightarrow{10}_{z}}^{−1} = f_{σ}(x, y)^{−1}p_{z}, \\ σ(p_{w}) & = σ\overrightarrow{01}_{w}p_{w}σ_{\overrightarrow{10}_{w}}^{−1} =: F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1})^{−1}p_{w}, \\ σ(p_{w}δ_{n}^{−1}) & = σ\overrightarrow{01}_{w}p_{w}δ_{n}^{−1}σ_{n\overrightarrow{10}_{w}}^{−1} = f_{σ}^{(n)}(ξ, η_{0}, η_{1}, …, η_{n−1})^{−1}p_{w}δ_{n}^{−1}. \end{matrix}

(A7) Lemma. (1) σ(δ_{n}) = σ_{n\overrightarrow{10}_{w}}δ_{n}σ_{\overrightarrow{10}_{w}}^{−1} = δ_{n}η_{0}^{ρ_{n}(σ)}.

(2) β_{n}(n\overrightarrow{10}_{w}) = \overrightarrow{10}_{z} (hence β_{n}(p_{w}δ_{n}^{−1}) = p_{z}).

(3) f_{σ}^{(n)}(ξ, η_{0}, η_{1}, …, η_{n−1}) = η_{0}^{ρ_{n}(σ)}F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1}).

(4) ψ_{n}^{(0)}(F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1})) = 0.

Proof. Calculations:

(1) It suffices to check the effect of both sides on w_{1}^{1/N} ∈ ℚ{{w_{1}}}:

σ(δ_{n})(w_{1}^{1/N}) = σ_{n\overrightarrow{10}_{w}}δ_{n}σ_{\overrightarrow{10}_{w}}^{−1}(w_{1}^{1/N}) = σ_{n\overrightarrow{10}_{w}}δ_{n}(w_{1}^{1/N}) = σ_{n\overrightarrow{10}_{w}}( \sqrt[N]{n}^{−1}(nw_{1})^{1/N})
= ζ_{N}^{−ρ_{n}(σ)} \sqrt[N]{n}^{−1}(nw_{1})^{1/N} = δ_{n}(ζ_{N}^{−ρ_{n}(σ)}w_{1}^{1/N}) = δ_{n}η_{0}^{ρ_{n}(σ)}(w_{1}^{1/N}).

Notice that η_{0}(w_{1}^{1/N}) = ζ_{N}^{−1}w_{1}^{1/N} in our convention.

(2) β_{n}^{∗}(z_{1}) = 1 − (1 − w_{1})^{n} = nw_{1}(1 − \frac{n − 1}{2}w_{1} + ⋯) ∈ ℚ\{\{w_{1}\}\}.

(3) Recalling the above definition of F_{σ} in (A6), one has

σ(p_{w}δ_{n}^{−1}) = F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1})^{−1}p_{w}(δ_{n}η_{0}^{ρ_{n}(σ)})^{−1}
= F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1})^{−1}η_{0}^{−ρ_{n}(σ)}p_{w}δ_{n}^{−1}
= (η_{0}^{ρ_{n}(σ)}F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1}))^{−1}p_{w}δ_{n}^{−1}.

Hence

f_{σ}(x, y) = β_{n}(η_{0}^{ρ_{n}(σ)}F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1}))
= y^{ρ_{n}(σ)}F_{σ}(x^{n}, y, xyx^{−1}, …, x^{n−1}yx^{−(n−1)}).

(4) Since ι_{n}(p_{w}) = p_{z}, it follows that

f_{σ}(x, y) = ι_{n}(F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1})) = F_{σ}(x, y, 1, …, 1).

Thus, if we put F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1}) ≡ ξ^{a}η_{0}^{b_{0}}η_{1}^{b_{1}} ⋯ η_{n−1}^{b_{n−1}} mod F̂′_{n+1}, we have a = b_{0} = 0 because f_{σ}(x, y) ∈ F̂′_{2}. □

Proof of Proposition (A1): By Lemma (A2) and Lemma (A7) (3),(4), one sees

Ψ_{n}^{(0)}(σ) = ψ_{n}^{(0)}(f_{σ}^{(n)}(ξ, η_{0}, η_{1}, …, η_{n−1}))
= ψ_{n}^{(0)}(η_{0}^{ρ_{n}(σ)}F_{σ}(ξ, η_{0}, η_{1}, …, η_{n−1})) = ρ_{n}(σ). □

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Department of Mathematics, Okayama University, Okayama 700-8530, Japan

E-mail address: h-naka@math.okayama-u.ac.jp

Department of Mathematics, Sophia University, Tokyo 102-8544, Japan

E-mail address: tsuno@mm.sophia.ac.jp