Introduction↩︎

The purpose of this paper is to introduce the triple quadratic residue symbol \([\frak{p}_1, \frak{p}_2, \frak{p}_3]\) for certain primes \(\frak{p}_i\)’s of real quadratic fields and investigate its properties. Our symbols is a triplication of quadratic residue symbol in real quadratic fields, and is also a generalization of Rédei’s triple symbol over the rational number field \(\mathbb{Q}\) to real quadratic fields.

The study of triple symbols in arithmetic goes back to the work of Rédei in 1939 ([1]), which intended to generalize the Legendre symbol and Gauss’ genus theory for quadratic fields. For certain prime numbers \(p_1, p_2\) and \(p_3\), Rédei’s triple symbol \([p_1, p_2, p_3] \in \{ \pm 1\}\) describes the decomposition law of \(p_3\) in the dihedral extension of degree 8 over \(\mathbb{Q}\), determined by \(p_1\) and \(p_2\). Some variants of the Rédei symbol have also been studied in [2], [3], [4] among others. In the late 1990s, Morishita interpreted the Rédei symbol as a mod 2 arithmetic analogue of the triple linking number (Milnor invariant) of a link \(L\) in the 3-sphere \(S^3\) ([5], [6]) from the viewpoint of arithmetic topology ([7]), especially from the analogy between Koch’s theorem on a group presentation of the Galois group \(\pi_1^{ pro-2}({\rm Spec}(\mathbb{Z}) \setminus S)\) of the maximal pro-2 extension over \(\mathbb{Q}\) unramified outside a finite set \(S\) of primes and Milnor’s theorem on a group presentation of a link group \(\pi_1(S^3 \setminus L)\), and introduced the multiple quadratic residue symbols \([p_1, \dots , p_r]\) for certain prime numbers \(p_i\)’s as mod 2 arithmetic analogues of Milnor invariants of a link ([8], [9], [10]). He also gave a description of \([p_1, \dots , p_r]\) in terms of the Massey product of the étale cohomology of \({\rm Spec}(\mathbb{Z}) \setminus \{(p_1), \dots, (p_r)\}\) ([10]).

Now it remains an important problem to generalize the multiple quadratic residue symbols in \(\mathbb{Q}\) to the multiple power residue symbols for primes in a number field. We may note that Turaev ([6]) extended Milnor’s theorem on a link group to an integral homology \(3\)-sphere \(M\) and defined Milnor’s invariants of a link in \(M\). Since the ideal class group is regarded as an analogue of the \(1\)st homology group of a \(3\)-manifold in arithmetic topology, it is natural to expect that we may extend Koch’s presentation of the pro-\(2\) Galois group over \(\mathbb{Q}\) to the Galois group \(G_{k, S}(l)\) of the maximal pro-\(l\) extension over a number field \(k\) unramified outside a finite set \(S\) of primes, when \(k\) has the (narrow) class number one and \(k\) contains the \(l\)-th roots of unity (\(l\) being a prime number), and that we may derive multiple \(l\)-th power residue symbols from \(G_{k, S}(l)\). However, there are a cohomological obstruction to determine a group presentation of \(G_{k, S}(l)\) and the difficulty arising from the unit group of \(k\), in order to define arithmetic Milnor invariants. So far there is the only case \(k = \mathbb{Q}(\sqrt{-3})\) and \(l=3\), for which triple cubic residue symbols are defined by using the Galois group \(G_{\mathbb{Q}(\sqrt{-3}), S}(3)\) ([11]). For other approaches to triple symbols, we refer to the recent works [12] and [13].

In this paper, we resolve the difficulty arising from the cohomological obstruction and the unit group for a real quadratic field \(k\) and determine a presentation of the Galois group \(G_S(2)\) of the maximal pro-2 extension of \(k\) unramified outside a finite set of finite primes \(S = \{\frak{p}_1, \dots \frak{p}_s\}\) and the infinite primes, when \(k\) has the trivial narrow class group. From this we are able to derive the triple quadratic residue symbol \([\frak{p}_1, \frak{p}_2, \frak{p}_3] = \pm1\) for certain finite primes of \(k\). To be precise, fix a prime number \(p \equiv 1\) mod 4 and let \(k\) be a real quadratic field \(\mathbb{Q}(\sqrt{p})\) whose narrow ideal class group is trivial, and let \(\varepsilon\) be the fundamental unit of \(k\). Let \(S = \{ \frak{p}_1, \dots, \frak{p}_s \}\) be a finite set of finite primes of \(k\) and let \(\infty_1, \infty_2\) denotes the infinite primes of \(k\), where \({\rm N}\frak{p}_i \equiv 1\) mod 4 and the corresponding embeddings \(\iota_{\infty_j} : k \hookrightarrow \mathbb{R}\) are defined by \(\iota_{\infty_1}(a + b\sqrt{p}) = a + b\sqrt{p}\), \(\iota_{\infty_2}(a + b\sqrt{p}) = a - b\sqrt{p}\) for \(a, b \in \mathbb{Q}\). In Section 1, we showed as our first result that the pro-2 Galois group \(G_S(2)\) has the following presentation \[\label{intro1} G_S(2) = \langle x_1, \cdots , x_s \mid x_1^{\mathrm{N}\mathfrak{p}_{1}-1} [ x_1, y_1] = \cdots = x_s^{\mathrm{N}\mathfrak{p}_{s}-1} [ x_s, y_s] =1, x_{\infty_2}^2 = 1 \rangle,\tag{1}\] where the word \(x_{i}\) represents a monodromy \(\tau_i\) over \(\mathfrak{p}_{i}\) in \(k_S(2)/k\) and \(y_{i}\) denotes the free pro-\(2\) word of \(x_{1}, \dots, x_{s}\) which represents a Frobenius automorphism \(\sigma_i\) over \(\mathfrak{p}_{i}\) in \(k_S(2)/k\), and \(x_{\infty_2}\) denotes the word which represents the monodromy over a real prime \(\infty_2\) (see Theorem 3). We note that the presentation 1 looks similar to that of a link group ([5], [6]) by the analogy between a monodromy (resp. Frobenius automorphism) over a prime and a meridian (resp. longitude) of a knot ([7]).

In Section 2, we define the triple quadratic residue symbol \([\frak{p}_1, \frak{p}_2, \frak{p}_3]\) using group presentation obtained in Section 1, and we also confirm that this definition is a generalization of the quadratic residue symbol. To be precise, let us consider the case that \(S = \{ \frak{p}_1, \frak{p}_2, \frak{p}_3 \}\) with \(\frak{p}_i = (\pi_i)\) such that \[\displaystyle{\left( \frac{\pi_i}{\pi_j} \right) = 1} \;{\rm for}\; 1 \leq i \neq j \leq 3 \;{\rm and}\; \displaystyle{\left( \frac{\varepsilon}{\pi_i} \right)} = 1 \;{\rm for}\;1 \leq i \leq 3,\] where \(\displaystyle{\left(\frac{\cdot}{\cdot} \right)}\) denotes the quadratic residue symbol in \(k\). Then, following the method by Morishita ([8], [9], [10]), we obtain the well-defined mod 2 Milnor invariant \(\mu_2(123)\) by using the presentation 1 and hence the triple quadratic residue symbol \[[\frak{p}_1, \frak{p}_2, \frak{p}_3] := (-1)^{\mu_2(123)}.\] By the definition of \(\mu_2(123)\), the triple symbol \([\frak{p}_1, \frak{p}_2, \frak{p}_3]\) describes the decomposition law of \(\frak{p}_3\) in a dihedral extension \(K\) of degree 8 over \(k\), unramified outside \(S\) and the infinite primes with ramification index of \(\frak{p}_i\) being 2. We call such an extension \(K\) a Rédei type \(D_8\)-extenision over \(k\). Actually we show that Rédei type extenision \(K\) exists uniquely for \(\{\frak{p}_1, \frak{p}_2\}\) (see Theorem 10).

In Section 3, we construct Rédei type \(D_8\)-extenision \(K\) under several conditions, following the method by Rédei ([14], [1]). To be precise, let \(k=\mathbb{Q}(\sqrt{p})\) be a real quadratic field and we assume \(p \equiv 5\;{\rm mod}\;8\). Let \(\frak{p}_1 = (\pi_1), \frak{p}_2 = (\pi_2)\) are prime ideals of \(\mathcal{O}_k\) such that \(\pi_i \equiv 1\;{\rm mod}\;4\mathcal{O}_k\) and \(\pi_i > 0\;(1 \leq i \leq 2).\) Then, there is a non-zero \(\mathcal{O}_k\) solution \((x, y, z)\) of \(x^2 - \pi_1 y^2 - \pi_2 z^2 = 0\) such that \({\rm gcd}(x, y, z) = 1\), \(y \in 2\mathcal{O}_k\). Furthermore, if \(x - y \equiv 1\; {\rm mod}\; 4\mathcal{O}_k\), then the Rédei type \(D_8\)-extenision \(K\) is given by \[K = k(\sqrt{\pi_1}, \sqrt{\pi_2}, \sqrt{x+y\sqrt{\pi_1}}).\] We also give some numerical examples of Rédei type \(D_8\)-extenisions over real quadratic fields. Moreover, we give the first example of a Borromean primes \(\frak{p}_1 = (\pi_1), \frak{p}_2 = (\pi_2), \frak{p}_3 = (\pi_3)\) in \(\mathbb{Q}(\sqrt{5})\), which satisfies \(\left(\frac{\pi_i}{\pi_j} \right) = 1\) (\(i \neq j\)) and \([\frak{p}_1, \frak{p}_2, \frak{p}_3] = -1\).

Finally, in Section 4, following [10], we give a cohomological interpretation of mod 2 Milnor invariant and the triple quadratic residue symbol from the perspective of Massey products.

Notations↩︎

For a set \(A\), \(\#A\) denotes the cardinality of \(A\).
For a finite algebraic number field \(F\), \({\mathcal{O}}_F\) denotes the ring of integers of \(F\), \(H_F\) denotes the ideal class group of \(F\), and \(H_F^+\) denotes the narrow ideal class group of \(F\).
For an ideal \(\frak{a}\) of \({\mathcal{O}}_F\), \({\rm N}\frak{a} := \#({\mathcal{O}}_k/\frak{a})\).
For a group \(G\) and \(x, y \in G\), \([x,y] := xyx^{-1}y^{-1}\).
For a topological group \(G\) and subsets \(X, Y\) of \(G\), \([X, Y]\) denotes the closed subgroup of \(G\) generated by \([x,y]\) for \(x \in X, y \in Y\).

1 Maximal pro-2 Galois groups of real quadratic fields with restricted ramification↩︎

In this section, we study a presentation of the Galois group \(G_S(2)\) of the maximal pro-2 extension over a real quadratic field \(k\), which is unramified outside a finite set \(S\) of primes of \(k\). For this, we apply theorems due to Höchsmann and Koch ([15]) and we assume that the narrow class group of \(k\) is trivial and a certain cohomological obstruction \(B_S\) vanishes, in order to give a precise minimal generators and their relations. We also give a condition for \(B_S = \{1\}\) in terms of the fundamental unit of \(k\).
Let \(k\) be a real quadratic field and \(\mathcal{O}_k\) be the ring of integers of \(k\). We recall that the unit group \(\mathcal{O}_k^{\times}\) is given by \[\mathcal{O}_k^{\times}=\{ \pm \varepsilon^n \mid n \in \mathbb{Z}\},\] where \(\varepsilon\) is the fundamental unit of \(k\). Let \(S\) be a finite set of finite primes of \(k\), \[S = \{\frak{p}_1, \cdots, \frak{p}_s\}, \;\#S = s.\] We assume that \(\mathfrak{p}_i\)’s are not lying over 2, that is, \({\mathrm N} {\mathfrak p}_i \equiv 1 \;\mathrm{mod} \; 2\) for each \(i\). Let \(\overline{S}\) be the union \[\overline{S} = S \cup S_{\mathbb{R}},\] where \(S_{\mathbb{R}} = \{\infty_1, \infty_2\}\) is the set of real primes of \(k\). For \(\frak{p}_i \in {S}\), let \(k_{\frak{p}_i}\) be the completion of \(k\) at \(\frak{p}_i\) and let \(\pi_i\) be a prime element of \(k_{\frak{p}_i}\). Each primes \(\frak{p} \in \overline{S}\) defines an embedding \(k\) to \(k_{\frak{p}}\), which we denote by \(\iota_\frak{p}\). In particular, for real primes, we denote by \(\iota_{\infty_1}\) the embedding into \(\mathbb{R}\) determined from the identity map and the conjugate map by \(\iota_{\infty_2}\). Let \(k_S(2)\) denote the maximal pro-\(2\) extension of \(k\), unramified outside \(\overline{S}\) and we set \(G_S(2) := \mathrm{Gal}(k_S(2)/k)\).

Now we describe the structure of the pro-\(2\) group \(G_S(2)\) in a certain unobstructed case. For this, we first recall a result due to Iwasawa on the local Galois group ([15]). We fix an algebraic closure \(\overline{k}_{\frak{p}_i}\) of \(k_{\frak{p}_i}\) and an embedding \(\bar{k} \hookrightarrow \overline{k}_{\frak{p}_i}\). Let \(k_{\frak{p}_i}(2)\) denote the maximal pro-\(2\) extension of \(k_{\frak{p}_i}\) in \(\overline{k}_{\frak{p}_i}\) and \(G_{k_{\frak{p}_i}}(2)\) denote the Galois group of \(k_{\frak{p}_i}(2)\) over \(k_{\frak{p}_i}\). In this case, the following identity is known: \[k_{\frak{p}_i}(2) = k_{\frak{p}_i}( \zeta_{2^n}, \sqrt[2^n]{\pi_i} \; \mid \; n \geq 1 ),\] where \(\zeta_{2^n}\) denotes a primitive \(2^n\)-th root of unity in \(\overline{k}\) such that \((\zeta_{2^a})^{2^b} = \zeta_{2^{a-b}}\) for all \(a \geq b\). The local Galois group \(G_{k_{\frak{p}_i}}(2)\) is topologically generated by the monodromy \(\tau_i\) and (a lift of) the Frobenius automorphism \(\sigma_i\) which are defined by \[\begin{array}{ll} \tau_i(\zeta_{2^n}) := \zeta_{2^n}, & \tau_i(\sqrt[2^n]{\pi_i}) := \zeta_{2^n}\sqrt[2^n]{\pi_i},\\ \sigma_i(\zeta_{2^n}) := \zeta_{2^n}^{{\mathrm N}\frak{p}_i},& \sigma_i(\sqrt[2^n]{\pi_i}) := \sqrt[2^n]{\pi_i}, \end{array}\] which are subject to the relation \[\label{relationfiniteprime} \tau_i^{{\mathrm N}\frak{p}_{i} -1}[\tau_i, \sigma_i] = 1.\tag{2}\] For \(\frak{p} \in S_{\mathbb{R}}\), we have \(k_{\frak{p}} = \mathbb{R}\), \(k_{\frak{p}}(2) = \mathbb{C}\), and \[\label{relationinfinite} G_{k_\frak{p}}(2) = \langle \;\tau \mid \tau^2=1 \;\rangle.\tag{3}\] For each \(\frak{p} \in \overline{S}\), the fixed embedding \(\bar{k} \hookrightarrow \overline{k}_{\mathfrak{p}}\) induces an embedding \(k_S(2) \hookrightarrow k_{\mathfrak{p}}(2)\), so a prime \(\mathfrak{P}_i\) of \(k_S(2)\) lying over \(\mathfrak{p}_i\) (\(1 \leq i \leq s\)). We denote by the same letters \(\tau_i\) and \(\sigma_i\) the images of \(\tau_i\) and \(\sigma_i\) under \(\varphi_{\mathfrak{p}}\), respectively, under the homomorphism \[\varphi_{\mathfrak{p}} : G_{k_{\mathfrak{p}}}(2) \longrightarrow G_S(2)\] induced by the embedding \(k_S(2) \hookrightarrow k_{\mathfrak{p}}(2)\). Then \(\tau_i\) is a topological generator of the inertia group of \(\mathfrak{P}_i\) and \(\sigma_i\) is a lift of the Frobenius automorphism of the maximal subextension of \(k_S(2)/k\) in which \(\mathfrak{P}_i\) is unramified. We call simply \(\tau_i\) a monodromy over \(\mathfrak{p}_i\) in \(k_S(2)/k\) and \(\sigma_i\) a Frobenius automorphism over \(\mathfrak{p}_i\) in \(k_S(2)/k\). We note that the restriction of \(\tau_i\) to the maximal Abelian subextension \(k_S(2)^{\rm ab}\) in \(k_S(2)/k\), denoted by \(\overline{\tau}_i := \tau_i|_{k_S(2)^{\rm ab}}\), is given by the Artin symbol \[\overline{\tau}_i = ( \eta_i, k_S(2)^{\rm ab}/k),\] for an idèle \(\eta_i\) of \(k\) whose \(\mathfrak{p}_i\)-component is a primitive \((\mathrm{N} \mathfrak{p}_i-1)\)-th root of unity in \(k_{\mathfrak{p}_i}^{\times}\) and all of the other components are 1. Now we assume that the \(2\)-class group \(H_k(2)\) of \(k\) is trivial. Then, by class field theory and Burnside’s basis theorem ([15]), the monodromies \(\tau_1, \dots , \tau_s, \tau_{\infty_1}, \tau_{\infty_2}\) generate topologically the global Galois group \(G_S(2)\). However, they may not be a minimal set of generators in general. In fact, Shafarevich’s theorem ([15]) tells us that the minimal number \(d(G_S(2))\) of generators of \(G_S(2)\) is given by \[\label{dGs} d(G_S(2)) = s + \dim_{\mathbb{F}_2} B_S,\tag{4}\] where the obstruction \(B_S\) is defined as follows: We define the subgroup \(V_S\) of \(k^{\times}\) by \[V_S := \{ a \in k^{\times} \mid a{\mathcal{O}}_k = \frak{a}^2, \; a \in (k_{\frak{p}}^{\times})^2 \; (\forall \frak{p} \in \overline{S}) \},\] where \(\mathfrak{a}\) is a fractional ideal of \({\mathcal{O}}_k\), and then \(B_S\) is defined by \[B_S := V_S/(k^{\times})^2,\] which is an elementary Abelian \(2\)-group. By Shafarevich’s formula 4 on \(d(G_S(2))\), \(2-\dim_{\mathbb{F}_2} B_S\) generators among \(\tau_i\)’s have to be removed in order to get a minimal generators of \(G_S(2)\).

Now we show a necessary and sufficient condition for \(B_S\) to vanish by using the fundamental unit \(\varepsilon\) of \(k\). For \(a \in k^{\times}\) and a finite prime \(\frak{p}\) of \(k\) such that \(2, a\) and \(\frak{p}\) are relatively prime, \(\left(\frac{a}{\frak{p}}\right)\) denotes the quadratic residue symbol in \(k\). For a real prime \(\infty\) of \(k\), \(\left(\frac{a}{\infty}\right)\) is defined by \(1\) (resp. \(-1\)) if \(\iota_\infty(a) > 0\) (resp. if \(\iota_\infty(a) < 0\)).

::: {#B_S=1 .prp} Proposition 1 (). (1) We assume that the ideal class group \(H_k\) of \(k\) is trivial. Then, the following conditions are equivalent.
(i) \(B_S = \{ 1 \}\).
(ii) There is a prime \(\frak{p} \in \overline{S}\) such that \(\displaystyle{\left(\frac{\varepsilon}{\frak{p}}\right)} = - 1\).
(2) We assume that the narrow ideal class group \(H_k^+\) of \(k\) is trivial. Then we have \(\left(\frac{\varepsilon}{\infty_2}\right) = -1\) and \(B_S = \{ 1 \}\). :::

By using Proposition 1, the minimal set of generators of \(G_S(2)\) can be determined as follows. Let \(\Phi(G_S(2))\) be the Frattini subgroup of \(G_S(2)\), namely, \(\Phi(G_S(2)) = G_S(2)^2[G_S(2), G_S(2)].\)

::: {#structure of G_S .prp} Proposition 2 (). Notations being as above, we assume that the ideal class group \(H_k\) of \(k\) is trivial, and there is a prime \(\frak{p} \in \overline{S} \setminus \{\infty_1\}\) such that \(\left(\frac{\varepsilon}{\frak{p}}\right) = - 1\). Then \(G_{S}(2)\) is topologically generated by the set of minimal generators \(\{\tau_1, \dots, \tau_{s}, \tau_{\infty_2}\}\) \(\setminus \{\tau\}\), where \(\tau\) is a monodromy corresponding to \(\frak{p}\). If the narrow ideal class group \(H_k^+\) of k is trivial, then we can take \(\tau = \tau_{\infty_2}\). :::

By Proposition 1 and Proposition 2, the following holds.

The following Proposition 4 will be used in Section 2.

::: {#xinfty in frattini .prp} Proposition 4 (). Notations being as above, we assume that the narrow ideal class group \(H_k^+\) of \(k\) is trivial. Then, the following conditions are equivalent.
(i) \(x_{\infty_2} \in \Phi(G_S(2))\).
(ii) \(\displaystyle{\left(\frac{\varepsilon}{\frak{p}_i}\right)} = 1\) for all finite primes \(\frak{p}_i \in S\). :::

2 Mod 2 arithmetic Milnor invariants and triple quadratic residue symbols in real quadratic fields↩︎

In this section, following Morishita’s work ([8], [9], [10]), we introduce mod 2 arithmetic Milnor invariants and triple quadratic residue symbols for certain primes of a real quadratic field, by using the presentation in Theorem 3 in Section 1.
Now, let us be back to the arithmetic situation in Section 1 and keep the same notations and assumptions there. Let \(k\) be a real quadratic field. In the following, we assume that the narrow ideal class group \(H_k^+\) of \(k\) is trivial. Let \(S = \{\mathfrak{p}_1, \cdots, \mathfrak{p}_s\}\) be a set of \(s\) distinct finite primes of \(k\), where \(\mathfrak{p}_i\)’s are not lying over 2, and let \(\overline{S} := S \cup S_{\mathbb{R}}\), when \(S_{\mathbb{R}}\) denotes the set of real primes of \(k\). We fix a prime element \(\pi_i\) of \(\frak{p}_i\), namely, \(\frak{p}_i = (\pi_i)\). By Theorem 3, the pro-2 Galois group \(G_S(2)\) has the following presentation: \[G_S(2) = \langle x_1, \cdots , x_s \mid x_1^{\mathrm{N}\mathfrak{p}_{1}-1} [ x_1, y_1] = \cdots = x_s^{\mathrm{N}\mathfrak{p}_s-1} [ x_s, y_s] =1, x_{\infty_2}^2 = 1 \rangle,\] where \(x_{i}\) denotes the word which represents a monodromy over \(\mathfrak{p}_{i}\) in \(k_S(2)/k\) and \(y_{i}\) denotes the free pro-\(2\) word of \(x_{1}, \dots, x_{s}\) which represents a Frobenius automorphism over \(\mathfrak{p}_{i}\) in \(k_S(2)/k\), and \(x_{\infty_2}\) denotes the word which represents a monodromy over a real prime \(\infty_2\).

Let \(F_S\) denote the free pro-2 group on the words \(x_1, \dots, x_s\). Let \(\mathbb{F}_2[[F_S]]\) be the complete group algebra of \(F_S\) over \(\mathbb{F}_2\) and let \(\varepsilon_{\mathbb{F}_2[[F_S]]} : \mathbb{F}_2[[F_S]] \longrightarrow \mathbb{F}_2\) be the augmentation homomorphism with the augmentation ideal \(I_{\mathbb{F}_2[[F_S]]} := {\rm Ker}(\varepsilon_{\mathbb{F}_2[[F_S]]})\). Let \(\mathbb{F}_2 \langle \langle X_1, \dots, X_s \rangle \rangle\) denote the formal power series algebra over \(\mathbb{F}_2\) in non-commutative variables \(X_1, \dots , X_s\). For a multiple index \(I=(i_1 \cdots i_n), 1\leq i_1,\dots ,i_n \leq s\), we set \(X_I = X_{i_1} \cdots X_{i_n}\). Let \[M : \mathbb{F}_2[[F_S]] \; \stackrel{\sim}{\longrightarrow} \; \mathbb{F}_2\langle\langle X_1,\dots ,X_s \rangle\rangle,\] be the mod \(2\) Magnus isomorphism defined by \[M(x_i) = 1+X_i,\enspace M(x_i^{-1}) = 1-X_i+X_i^2- \cdots .\] Then, for \(f \in F_S\), \(M(f)\) has the form (called Mod \(2\) Magnus expansion) \[M(f) = 1 + \sum_{|I| \geq 1} \mu_2(I;f) X_I,\] where \(\mu_2(I;f) \in \mathbb{F}_2\) are the mod \(2\) Magnus coefficients. In terms of the pro-2 Fox free derivatives \(\frac{\partial}{\partial x_i} : \mathbb{Z}_2[[F_S]] \longrightarrow \mathbb{Z}_2[[F_S]]\) over \(\mathbb{Z}_2\), we can obtain for \(I=(i_1 \cdots i_n)\) \[\mu_2(I;f) = \varepsilon_{{\mathbb{F}_2[[F_S]]}}\left(\frac{\partial^n f}{\partial x_{i_1} \cdots \partial x_{i_n}} \;{\rm mod}\; 2 \right),\] where mod \(2\) : \(\mathbb{Z}_2[[F_S]] \longrightarrow \mathbb{F}_2[[F_S]]\) is the reduction mod \(2\).

Here, we recall basic properties of mod \(2\) Magnus coefficients.

We note that the set \(\{F_S^{(q)}\}\) forms a fundamental system of neighbourhoods of 1 in \(F_S\) and that \(F_S^{(q)}/F_S^{(q+1)}\) is central in \(F_S/F_S^{(q+1)}\) and \((F_S^{(q)})^2 \subset F_S^{(2q)}\).

Now let \(I = (i_1 \cdots i_n)\) be a multi-index with \(1\leq i_1, \dots , i_n \leq s\). Let \(I' := (i_1 \cdots i_{n-1})\). Then the mod \(2\) arithmetic Milnor number \(\mu_2(I)\) for \(I\) is defined by \[\mu_2(I) := \mu_2(I'; y_{i_n}).\] Here we set \(\mu_2(I) := 0\) if \(|I| = 1\). The mod \(2\) arithmetic Milnor numbers \(\mu_2(I)\) are not invariants determined by the Galois group \(G_S(2)\). However, we can show in the following that \(\mu_2(I)\) are invariant of \(G_S(2)\) if \(|I| = 2\) or \(3\).

::: {#millnor residue .prp} Proposition 6 (). Notations being as above, let \(i, j\) be indices between \(1\) and \(s\). When \(i=j\), we have \(\mu_2(i i) = 0\). When \(i \neq j\), we have \[(-1)^{\mu_2(i j)} = \left(\frac{\pi_i}{\pi_j}\right).\] :::

Next, we show that mod \(2\) arithmetic triple Milnor numbers are invariants determined by \(G_S(2)\).

By Theorem 7, we call \(\mu_2(I)\) the mod 2 arithmetic triple Milnor invariant of \(S\) for the multi-index \(|I| = 3\), when \(\left(\frac{\varepsilon}{\frak{p}_i}\right) = 1\) for \(\frak{p}_i \in S\), \(\mu_2(J) = 0\) for any multi-index \(J\) of length \(\leq 2\) and \({\rm N}\frak{p}_i \equiv 1\) mod \(4\) for \(\frak{p}_i \in S\).

A meaning of the mod 2 arithmetic triple Milnor invariants is given as follows. Let \(\overline{S} = S \cup S_{\mathbb{R}} = \{ \frak{p}_1, \frak{p}_2 , \frak{p}_3\} \cup \{\infty_1,\infty_2\}\) be a finite set of primes of \(k\), where \({\rm N}\frak{p}_i \equiv 1 \;{\rm mod} \;4\) (\(1 \leq i \leq 3\)) and \(\infty_1, \infty_2\) are real primes. We assume that \(\mu_2(ij) = 0\) for any multi-index of length \(2\). By Proposition 6, it means \(\left(\frac{\pi_i}{\pi_j}\right)=1\) for \(i \neq j\). Let \(N_3(\mathbb{F}_2)\) denote the group of \(3\) by \(3\) upper-triangular unipotent matrices over \(\mathbb{F}_2\). We note that \(N_3(\mathbb{F}_2)\) is isomorphic to the dihedral group \(D_8 = \langle s, t \; | \; s^2 = t^4 = 1, sts^{-1} = t^{-1} \rangle\) of order \(8\) by the correspondence \[s \mapsto \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right), \;\; t \mapsto \left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right).\] We define the map \[\rho : F_{S} \longrightarrow N_3(\mathbb{F}_2)\] by \[\rho(f) := \left( \begin{array}{ccc} 1 & \mu_2((1);f) & \mu_2((1 2); f)\\ 0 & 1 & \mu_2((2); f)\\ 0 & 0 & 1 \\ \end{array} \right)\] for \(f \in F_S\). By Lemma 5 (i), we see \(\rho\) is a group homomorphism. Let \(S' = \{\frak{p}_1, \frak{p}_2\}\) and let \(\overline{S'} = S' \cup S'_{\mathbb{R}}\) be the subset of \(S\) and \(\overline{S}\).

Suggested by Proposition 6, we may define the triple quadratic residue symbol \([\frak{p}_1, \frak{p}_2, \frak{p}_3]\) as follows.

3 Rédei type \(D_8\)-extensions over real quadratic fields↩︎

In this section, we give examples of Rédei type \(D_8\)-extensions of \(k\) for \(\{\frak{p_1}, \frak{p}_2\}\) in Theorem 10. We keep the same notations and assumptions as in the Section 2. Namely, let \(k\) be a real quadratic field whose narrow class number is one. By Gauss’ genus theory ([17]), there is a prime number \(p = 2\) or \(p \equiv 1\) mod \(4\) such that \(k = \mathbb{Q}(\sqrt{p})\). We assume that \(p \equiv 1\) mod \(4\). Let \(\frak{p}_1 = (\pi_1), \frak{p}_2 = (\pi_2), \frak{p}_3 = (\pi_3)\) be distinct finite primes of \(k\) such that \({\rm N}\frak{p}_i \equiv 1\) mod 4, \(\left(\frac{\varepsilon}{\frak{p}_i}\right) = 1\) (\(1 \leq i \leq 3\)) and \(\left(\frac{\pi_i}{\pi_j}\right) = 1\) (\(1 \leq i \neq j \leq 3\)). Let \(p_i\) denote the rational prime lying below \(\frak{p}_i\).

Next, we consider the case that \(p_1\) or \(p_2\) is decomposed in \(k/\mathbb{Q}\). To deal with this case, we recall some lemmas regarding ramification.

Using Lemma 15, we obtain the following lemma.

Here is a numerical example of Theorem 18.

Next, we give other numerical examples, which are not contained in the cases of Proposition 14 and Theorem 18.

4 Massey products in Galois cohomology↩︎

In this section, we interpret our mod 2 arithmetic Milnor invariants and triple quadratic residue symbols in terms of the Massey product in Galois cohomology. Our theorem is seen as a generalization of the known relation between the cup product and the quadratic residue symbol to the triple case, and also a generalization of the previous result by Morishita ([10]) in the case of the rational number field to the real quadratic fields. It may be regarded as a mod 2 arithmetic analogue of the corresponding topological result due to Turaev ([6]).
Let \(G\) be a pro-\(2\) group and let \(A\) be a commutative ring with identity on which \(G\) acts trivally. Let \(C^j(G,A)\) be the \(A\)-module of inhomogeneous \(j\)-cochains \((j \geq 0)\) of \(G\) with coefficients in \(A\) and we consider the differential graded algebra \((C^{\bullet}(G,A), d)\), where the product structure on \(C^{\bullet}(G,A) = \bigoplus_{j\geq 0} C^j(G,A)\) is given by the cup product and the differential \(d\) is the coboundary operator. Then we have the cohomology \(H^*(G,A) = H^*(C^{\bullet}(G,A))\) of the pro-\(2\) group \(G\) with coefficients in \(A\). In the following, we consider Massey products in \(H^1(G,A)\). For the sign convention, we follow [19]. Let \(\chi_1,\dots,\chi_n \in H^1(G,A)\) \((n\geq 2)\). An \(n\)-th Massey product \(\langle \chi_1,\dots,\chi_n \rangle\) is said to be defined if there is an array \[\Omega = \{\omega_{ij} \in C^1(G,A) \; | \; 1 \leq i < j \leq n+1, (i,j) \neq (1,n+1) \}\] such that \[\left\{ \begin{array}{l} [\omega_{i,i+1}] = \chi_i \;\; (1\leq i \leq n),\\ \displaystyle{d\omega_{ij} = \sum_{a=i+1}^{j-1} \omega_{ia}\cup \omega_{aj}} \;\; (j \neq i+1). \end{array} \right.\] Such an array \(\Omega\) is called a defining system for \(\langle \chi_1,\dots, \chi_n \rangle\). The value of \(\langle \chi_1,\dots, \chi_n \rangle\) relative to \(\Omega\), denoted by \(\langle \chi_1,\dots, \chi_n \rangle_{\Omega}\), is defined by the cohomology class represented by the \(2\)-cocycle \[\sum_{a=2}^n \omega_{1a} \cup \omega_{a,n+1}.\] We define the Massey product \(\langle \chi_1,\dots,\chi_n \rangle\) to be the subset of \(H^2(G,A)\) consisting of elements \(\langle \chi_1,\dots, \chi_n \rangle_{\Omega}\) for some defining system \(\Omega\). By convention, \(\langle \chi \rangle = 0\). We recall the following basic fact (cf. [20]).

Next, we recall a relation between Massey products and Magnus coefficients. Suppose that \(G\) is a finitely generated pro-\(2\) group with a minimal presentation \[1 \longrightarrow N \longrightarrow F \stackrel{\psi}{\longrightarrow} G \longrightarrow 1,\] where \(F\) is a free pro-\(2\) group on generators \(x_1, \dots, x_n\) with \(n = {\rm dim}_{\mathbb{F}_2} H^1(G, \mathbb{F}_2)\). We set \(\tau_i := \psi(x_i) \;(1\leq i \leq n)\). We assume that \(\psi\) induces the isomorphism \(F/\Phi(F) \simeq G/\Phi(G)\) so that \(\psi\) induces the isomorphism \(\psi^* : H^1(G, \mathbb{F}_2) \simeq H^1(F, \mathbb{F}_2)\). We let \[{\rm tra} : H^1(N, \mathbb{F}_2)^{G} \rightarrow H^2(G, \mathbb{F}_2)\] be the transgression defined as follows. For \(a \in H^1(N, \mathbb{F}_2)^{G}\), choose a 1-cochain \(b \in C^1(F, \mathbb{F}_2)\) such that \(b|_{N} = a\). Since the value \(db(f_1,f_2)\), \(f_i \in F\), depends only on the cosets \(f_i\) mod \(N\), \(db\) defines a 2-cocyle \(c\) of \(G\). Then \({\rm tra}(a)\) is defined by the class of \(c\). By the Hochschild-Serre spectral sequence, \({\rm tra}\) is an isomorphism and so we have the dual isomorphism, called the Hopf isomorphism, \[{\rm tra}^{\vee} : H_2(G, \mathbb{F}_2) \stackrel{\sim}{\rightarrow} H_1(N, \mathbb{F}_2)_{G} = N/N^2[N, F].\] Then we have the following proposition (cf. [21]). The proof goes in the same manner as in [10].

Let us be back in our arithmetic situation and keep the same notations as in Section 2. So let \(k\) be the real quadratic field and let \(S := \{ \frak{p}_1, \frak{p}_2, \frak{p}_3\}\) be a set of all distinct primes where \({\rm N}\frak{p}_i \equiv 1\) mod \(4\) and \(\left(\frac{\varepsilon}{\frak{p}_i}\right) = 1\). Let \(\overline{S} := S \cup S_{\mathbb{R}}\) where \(S_{\mathbb{R}} = \{\infty_1, \infty_2\}\) be the set of real primes of \(k\). By Theorem 3, we have the following minimal presentation of the Galois group \(G_S(2)\) of maximal pro-\(2\) extension over \(k\) unramified outside \(\overline{S}\) \[\begin{align} G_S(2) &= \left \langle x_1, x_2 , x_3 \, \middle | \begin{array}{c} x_1^{\mathrm{N}\frak{p}_{1}-1} [ x_1, y_1] = x_2^{\mathrm{N}\frak{p}_{2}-1} [ x_2, y_2] = x_3^{\mathrm{N}\frak{p}_3-1} [ x_3, y_3] =1\\ \\ x_{\infty_2}^2 = 1 \end{array} \right \rangle \\ &= F_S/N_S. \end{align}\] Here \(x_i\) denotes the word representing a monodromy \(\tau_i\) over \(\frak{p}_i\) in \(k_S(2)/k\) \((1\leq i \leq 3)\) and \(F_S\) denotes the free pro-\(2\) group on \(x_1, x_2, x_3\). The pro-\(2\) word \(y_i\) represents a Frobenius automorphism \(\sigma_i\) over \(\frak{p}_i\) in \(k_S(2)/k\) and \(N_S\) denotes the closed subgroup of \(F_S\) generated normally by \(x_{i}^{{\rm N}\frak{p}_{i} -1}[x_{i},y_{i}]\) for \(1\leq i \leq 3\) and \(x_{\infty_2}^2\). We set \(\delta_i := ({\rm tra}^{\vee})^{-1}(x_i^{{\rm N}\frak{p}_i -1}[x_i,y_i])\), where \({\rm tra}^{\vee} : H_2(G_S(2),\mathbb{F}_2) \stackrel{\sim}{\rightarrow} N_S/N_S^2[N_S,F_S]\) is the Hopf isomorphism. Let \(\chi_1, \chi_2, \chi_3 \in H^1(G_S(2),\mathbb{F}_2)\) be the Kronecker dual to the monodromies \(\tau_1, \tau_2, \tau_3\), namely, \(\chi_i(\tau_j) = \delta_{i,j}\).

We assume that \[\langle \chi_i, \chi_j \rangle = 0 \;\; (1\leq i \neq j \leq 3),\] which is equivalent to the condition \[\left( \frac{\pi_i}{\pi_j} \right) = 1 \;\; (1\leq i \neq j \leq 3)\] by Proposition 7, and the mod \(2\) Milnor invariants \(\mu_2(abc)\) (\(\{a,b,c\} = \{1,2,3\}\)) are well defined. By the definition of Massey products and Lemma 23, there are 1-cochains \(\omega_{13}, \omega_{24} \in C^1(G_S(2), \mathbb{F}_2)\) such that \[\langle \chi_1, \chi_2 \rangle = d\omega_{13}, \; \langle \chi_2, \chi_3 \rangle = d\omega_{24},\] and we have the triple Massey product \(\langle \chi_1, \chi_2, \chi_3 \rangle\) defined by \[\langle \chi_1, \chi_2, \chi_3 \rangle = [\chi_1 \cup \omega_{24} + \omega_{13} \cup \chi_3].\]

Acknowledgements↩︎

I would like to express my sincere gratitude to my supervisor, Professor Masanori Morishita, for suggesting the problem studied in this paper and for his guidance and cooperation in carrying out this research. I am grateful to Professors Yasushi Mizusawa, Dohyeong Kim and Doctor Sosuke Sasaki for useful communications. I would like to thank Yuki Ishida and Dingchuan Zheng for useful discussions and suggestions.

References↩︎