Introduction↩︎

Context↩︎

Let \(p\) be an odd prime and \(K\) be a number field. A prime \({\mathfrak q}\) of \(K\) is said to be tame if \(N({\mathfrak q})\equiv 1\pmod p\). For a set of finite tame primes \(S\) and \(T\) a set of primes disjoint from \(S\), we denote by \(G_{K,S}\) (resp.\(G_{K,S}^T\)), the Galois group of the maximal pro-\(p\) extension of \(K\) unramified outside \(S\) (and resp.totally split in \(T\), with \(T\) a set of primes disjoint from \(S\)). For \(K\coloneq \mathbb{Q}\), we use the notation \(G_S^T\).

Following the work of Golod and Shafarevich on class field towers, the structure of the groups \(G_{K,S}^T\) has become a central topic in Galois theory. Using their famous result, several infinite (indeed nonanalytic) groups have been constructed as \(G_{K,S}^T\), with various arithmetic applications ([1]–[3]). However, despite the abundance of such examples, relatively little is known about their structure. By class field theory, the groups \(G_{K,S}^T\) are FAB, i.e.every open subgroup has finite abelianization. The celebrated tame Fontaine–Mazur conjecture (see [4]) also predicts that every \(p\)-adic analytic quotient of \(G_{K,S}^T\) is finite. This highlights the importance of understanding such groups in the context of Galois representations.

Koch was the first to determine a complete presentation for the groups \(G_S\) (see [5], [6]), which opened many directions for further investigation. Building on these results and the concept of strongly free sequences and mild groups introduced by Anick [7], [8], Labute [9] was able to construct several groups \(G_{S}\) of cohomological dimension \(2\). He showed [9] that every finite set of tame primes \(S\) can be extended to a tame set \(\smash{\widetilde{S}}\) of size \(2\cdot |S|\) such that \(G_{\smash{\widetilde{S}}}\) is of cohomological dimension \(2\). For the case \(p=2\), we refer to [10], [11]. Later, Schmidt [12]–[14] interpreted these results in terms of the étale homotopy type of \(\mathrm{Spec}({\mathcal{O}}_{K,S})\), and showed that, for the sets \(S\) considered by Labute, \(\mathrm{Spec}({\mathcal{O}}_{K,S})\) is a \(K(\pi,1)\) space for \(p\) in the context of arbitrary number fields. We refer to [15] for an historical survey. For other applications related to mild groups in arithmetic, we refer to [16]–[19]. Currently, it is still unknown whether there exists a group \(G_{K,S}^T\) of finite cohomological dimension greater than \(2\).

Statement of the results↩︎

We assume throughout the paper that \(K\) does not contain \(\zeta_p\), a primitive root of unity. We denote by \({\rm cl}(K)\) the class group of \(K\) and \(d_K'\coloneq r_K+d_p{\rm cl}(K)\), where \(r_K\) is the torsion-free rank of \(\mathcal{O}_K^\times\) and \(d_p{\rm cl}(K)\) denotes the \(p\)-rank of the class group. We fix \(T_K\) a minimal set of primes generating the \(p\)-part of the class group \({\rm cl}(K)\). Note that \(T_K\) always exists using the Chebotarev Density theorem and \(|T_K|=d_p{\rm cl}(K)\). For instance, if \(p\) is large enough or \(K=\mathbb{Q}\), then \(T_K=\emptyset\). Let \(S_1\) and \(S_2\) be two sets of primes. We introduce \[\begin{align} V_{S_1}^{S_2}(K)\coloneq \big\{a\in K^{\times }\mid a\in K_{\mathfrak q}^{\times p}\text{ for }{\mathfrak q}\in S_1\text{ and }p\mid \nu_\mathfrak{t}(a)\text{ for }\mathfrak{t}\not\in S_2\big\}/K^{\times p}, \end{align}\] with \(\nu_\mathfrak{t}\) the normalized valuation associated to \(\mathfrak{t}\). This allows us to define the Koch-module \({{\textrm{\rurm{B}}}}_{S_1}^{S_2}(K)\) as the dual of \(V_{S_1}^{S_2}(K)\).

We say that \(S'\) is a Koch set if it is tame, disjoint from \(T_K\), and satisfies \((a)\) \({{\textrm{\rurm{B}}}}_{S'}^{T_K}(K)=1\), and \((b)\) \(G_{K,S'}^{T_K}=1\). A finite tame set \(S\) admits a Koch set if \(S\) contains a Koch set \(S'\) and is disjoint from \(T_K\). We show in Lemma [lem:Existence32of32Koch32Tuples] that a set \(S\) can always be extended to contain a Koch set. Furthermore, Proposition [prop:Probabilty32for32existence32of32Koch32set] implies that for \(S\) large enough one can expect \(S\) to already admit a Koch set with high probability. If \(S\) admits a Koch set, then an analog of the presentation of \(G_S\) given by Koch exists for \(G_{K,S}^{T_K}\). This was observed by Liu in [20]. We call these presentations ‘’extended’’  Koch-type presentations. For a precise definition, we refer to Section 2.2. A crucial piece of data of such a presentation is a linking type function denoted \(\mu_S^{T_K}\). Note that when \(K=\mathbb{Q}\), the group \(G_S\) has a Koch-type presentation and \(\mu_S\) is the usual linking map as considered by Labute [9] and many others (see for example [13], [19], [21]). Using an argument developed by Maire and Sankara in [22], we positively answer a question posed by Labute, see Theorem [thm:Realization32of32extended32linking32data], and extend it in our context. It roughly states, that every function \(\mu\) with the right domain can be realized as \(\mu_S^T\). Theorem [thm:Realization32of32extended32linking32data], coupled with well-chosen weighted Zassenhaus filtrations, is fundamental in our work. We state our main result.

A notable feature of Theorem 1 is that the size of \(S_1\) depends only on \(K\), and is completely independent of the size of \(S_0\). The careful choices on weighted Zassenhaus filtrations used simultaneously with Theorem [thm:Realization32of32extended32linking32data], and combined with arguments from Right Angled Algebras developed by the second author in [23] (see also Remark [rem:sharper32result32first32theorem]), are essential in our proof. Previously, Forré [11] used judicious choices of filtrations to infer results on mildness for the case \(p=2\). Gärtner [24] also noted that mildness of a presentation may depend on the choice of the filtration.

As a direct consequence, for the case \(K=\mathbb{Q}\), we refine [9].

A similar approach allows us to study some cohomological inverse problems in the context of restricted ramification with splitting. We refer to Section 4. Namely, we realize some quadratic algebras as cohomology algebras of groups \(G_{K,S}^T\) for some well chosen \(S\) and \(T\). For this purpose, we use graph theory. It has been extensively used to study which pro-\(p\) groups arise as maximal pro-\(p\) Galois groups of fields. Snopce and Zalesskii [25], together with Cassella and Quadrelli [26], characterized which pro-\(p\) RAAGs are maximal pro-\(p\) Galois groups. They resolved several conjectures in this class. For further work, see [23], [27]–[29].

Some numerical examples↩︎

The results and proofs of Theorem 1, and Section 4 can be made computationally effective to produce concrete examples. We used OSCAR (see [30]) for our computations. The source code and a Jupyter Notebook, containing all the examples, can be found on the first author’s GitHub under [31].

For example, by virtue of Theorem 1 we show in Subsection 5.1 that the group \(G_{S}\), for \(p=3\) and the set \(S\coloneq \{7,13,19,37,10639,826093\}\), is mild with respect to a weighted Zassenhaus filtration and therefore of cohomological dimension \(2\). We give further examples of this behavior also in the case of more general number fields in Section 5.1.

Furthermore, Subsection 5.2 produces examples of groups \(G_S\) whose \(\mathbb{F}_p\)-cohomology ring \(H^\bullet(G_S)\) comes from graph theory. Let \({\mathcal{A}}\) be the graded algebra over \(\mathbb{F}_3\) presented by four generators \(\{X_1,X_2,X_3,X_4\}\) and \(12\) relations: \[\{X_i^2, \quad X_uX_v+X_vX_u,\quad X_1X_3, \quad X_2X_4, \quad 1\leq i \leq 4, \quad 1\leq u< v\leq 4\}.\] Then for \(p=3\) and \(S\coloneq \{7,13,181,5563\}\) we have \(H^\bullet(G_{S})\simeq {\mathcal{A}}\). Subsection 5.3 gives more examples for other number fields and for certain quadratic algebras not coming from graph theory.

Organization of the paper↩︎

The first section introduces all of the group-theoretical arguments required for our results. In particular, Subsection 1.4 introduces the notion of \(({\mathbf{X}},e)\)-RAAGs, which are not in general RAAGs, but nevertheless have the same \(({\mathbf{X}},e)\)-filtration as RAAGs associated to certain graphs. The second section introduces all arithmetic arguments needed in this paper. In particular, it answers a question of Labute by proving Theorem [thm:Realization32of32extended32linking32data], using an argument from [22]. The third section is devoted to the proof of Theorem 1. The main group theoretical argument relies on suitable choices of \(({\mathbf{X}},e)\)-filtrations, which yield \(({\mathbf{X}},e)\)-RAAGs whose underlying graphs are triangle-free. The fourth section applies graph theory to the cohomological inverse problem. The last section focuses on numerical examples.

1 Group theoretic and algebraic preliminaries↩︎

In this section we introduce general group-theoretic tools and techniques that we need to prove most of our results. For general references, we refer to [16], [17], [23], [32], [33].

1.1 Filtered and graded (Lie)-algebras↩︎

Set an integer \(d\) and a finite set \({\mathbf{X}}\coloneq \{x_1,\dots, x_d\}\). We denote by \(\mathbb{F}_p\langle\!\langle X_1,..,X_d\rangle\!\rangle\) the algebra of noncommutative series in the variables \(X_1,\dots, X_d\) over \(\mathbb{F}_p\). A basis of the topological \(\mathbb{F}_p\)-vector space \(\mathbb{F}_p \langle\!\langle X_1,\dots, X_d\rangle\!\rangle\) is given by monomials \(X^{\alpha}\coloneq \smash{X_{i_{\alpha_1}}^{\alpha_1}\dots X_{i_{\alpha_n}}^{\alpha_n} }\), where \(\alpha\) is an \(n\)-tuple and \(i_{\alpha_k}\) is an element in \(\{1,\dots,d\}\). Consequently, every element \(x\) in \(\mathbb{F}_p \langle\!\langle X_1,\dots, X_d\rangle\!\rangle\) can be written as an infinite sum \(x\coloneq \sum_{\alpha}{a_\alpha X^\alpha}\), with \(a_{\alpha}\in \mathbb{F}_p\).

For a given \(e\coloneq (e_1,\dots, e_d)\in \mathbb{N}^d\) we denote by \(E_e({\mathbf{X}})\) the set \(\mathbb{F}_p\langle\!\langle X_1,..,X_d\rangle\!\rangle\) endowed with the filtration \(\{E_{e,n}({\mathbf{X}})\}_{n\in \mathbb{N}}\) induced by the following degree function: \[\begin{align} \omega_e \Big({\sum}_{\alpha} a_\alpha \cdot X^\alpha \Big) \coloneq \min_{\alpha} \{ \alpha_1 e_{i_{\alpha_1}}+\dots +\alpha_n e_{i_{\alpha_n}} \}. \end{align}\] We define \({\mathcal{E}}_e({\mathbf{X}})\coloneq \bigoplus_{n\in \mathbb{N}} {\mathcal{E}}_{e,n}({\mathbf{X}})\), with \({\mathcal{E}}_{e,n}({\mathbf{X}})=E_{e,n}({\mathbf{X}})/E_{e,n+1}({\mathbf{X}})\). Note that \({\mathcal{E}}_e({\mathbf{X}})\) is isomorphic to the (free) graded algebra \(\mathbb{F}_p \langle X_1,\dots ,X_d\rangle\), where every \(X_i\) is of degree \(e_i\). If \(x\) is a nonzero element in \(E_e({\mathbf{X}})\), we define its initial form \(\overline{x}\) as the image of \(x\) in \[{\mathcal{E}}_{e,\omega_e(x)}({\mathbf{X}})=E_{e,\omega_e(x)}({\mathbf{X}})/E_{e, \omega_e(x)+1}({\mathbf{X}}).\] We fix an ordering on the set \(\{X_i\}_{i=1}^d\). We say that \(X^{\alpha}<X^{\beta}\), if \(\omega_e(X^{\alpha})>\omega_e(X^\beta)\) and if we have equality, we use the lexicographic order. Define \(\widehat{x}\coloneq \max_{a_\alpha \neq 0}\{X^{\alpha}\}\).

Let \(I\) be a closed two-sided ideal of \(E_e({\mathbf{X}})\). We endow \(I\) with the filtration \(\{I_n\coloneq I\cap E_{e,n}({\mathbf{X}})\}_{n\in \mathbb{N}}\). The algebra \(A\coloneq E_e({\mathbf{X}})/I\) is endowed with a filtration that we call the quotient filtration (of \(E_e({\mathbf{X}})\) by \(I\), see [32]). We denote it by \(\{A_n\}_{n\in \mathbb{N}}\). We define \({\rm Grad}(A)\coloneq \bigoplus_{n\in \mathbb{N}} {\rm Grad}_n(A)\), where \({\rm Grad}_n(A)\coloneq A_n/A_{n+1}\). We also introduce the Hilbert series of \(A\) (resp.of a graded algebra \({\mathcal{A}}\coloneq {\mathcal{E}}_e({\mathbf{X}})/\mathcal{J}=\bigoplus_{n\in \mathbb{N}}{\mathcal{A}}_n\), for some graded ideal \({\mathcal{J}}\)) by: \[A(t)\coloneq \sum_{n\in \mathbb{N}}\dim_{\mathbb{F}_p}({\rm Grad}_n(A))t^n, \quad \text{and} \quad {\mathcal{A}}(t)\coloneq \sum_{n\in \mathbb{N}}\dim_{\mathbb{F}_p}({\mathcal{A}}_n)t^n.\]

1.2 Magnus’s isomorphism and applications in group theory↩︎

We assume that \(G\) is a finitely presented pro-\(p\) group presented by \(d\) generators \(\{\tau_x|\quad x\in {\mathbf{X}}\coloneq \{x_1,\dots, x_d\}\}\) and \(r\) relations \(\{l_1,\dots, l_r\}\). We denote this presentation \({\mathcal{P}}\). Let \(F({\mathbf{X}})\) be the free pro-\(p\) group on \(\{\tau_x, x\in {\mathbf{X}}\}\) and \(R\) be its closed normal subgroup on \(\{l_1,\dots, l_r\}\). We have an exact sequence: \[1\to R \to F({\mathbf{X}}) \to G \to 1.\]

Magnus (see for instance [32]) constructed an injective group homomorphism \(\varPsi_{F({\mathbf{X}})}:F({\mathbf{X}})\to \mathbb{F}_p\langle\!\langle X_1,...,X_d\rangle\!\rangle^{\times}\) induced by \(\tau_{x_i}\mapsto 1+X_i\). This induces a filtration on \(F({\mathbf{X}})\), which we call \(({\mathbf{X}}, e)\)-filtration, defined by:\[F_{e,n}({\mathbf{X}})\coloneq \{f\in F({\mathbf{X}}), \quad \varPsi_{F({\mathbf{X}})}(f)-1\in E_{e,n}({\mathbf{X}})\}.\] We also define \({\mathcal{L}}_e({\mathbf{X}})\) the free graded \(p\)-restricted Lie-algebra on \(\{X_1,\dots, X_d\}\) over \(\mathbb{F}_p\), where every \(X_i\) is endowed with weight \(e_i\). Magnus isomorphism [32] also allows us to construct an isomorphism \[{\rm Grad}(\varPsi_{F({\mathbf{X}})})\colon \bigoplus_{n\in \mathbb{N}} F_{e,n}({\mathbf{X}})/F_{e,n+1}({\mathbf{X}}) \to {\mathcal{L}}_e({\mathbf{X}}).\]

We write \(I_e(R)\) for the closed two-sided ideal of \(E_e({\mathbf{X}})\) generated by \(\{ \varPsi_{F({\mathbf{X}})}(l)-1\}_{l\in R}\). We set  \(E_e({\mathbf{X}},G)\coloneq E_e({\mathbf{X}})/I_e(R)\), equipped with the quotient filtration \(\{E_{e,n}({\mathbf{X}},G)\}_{n\in \mathbb{N}}\). The map \(\varPsi_{F({\mathbf{X}})}\) induces \(\varPsi_{{\mathbf{X}},G}\colon G \to E_e({\mathbf{X}},G)^{\times}\) mapping \(\tau_{x_i}\) to the class of \(1+X_i\). We denote by \({\mathcal{I}}_e(R)\) the ideal of \({\mathcal{E}}_e({\mathbf{X}})\) isomorphic to \(\bigoplus_{n\in \mathbb{N}}I_{e,n}(R)/I_{e,n+1}(R)\). This allows us to define: \[E_e({\mathbf{X}},G)\coloneq E_e({\mathbf{X}})/I_e(R), \quad \text{and} \quad {\mathcal{E}}_e({\mathbf{X}},G)\coloneq {\mathcal{E}}_e({\mathbf{X}})/{\mathcal{I}}_e(R) \coloneq \bigoplus_{n\in \mathbb{N}} {\mathcal{E}}_{e,n}({\mathbf{X}}, G).\] We define, for every integer \(n\), the \(({\mathbf{X}}, e)\)-filtration of \(G\) by \[G_{e,n}({\mathbf{X}})\coloneq \{g\in G, \varPsi_{{\mathbf{X}},G}(g)-1\in E_{e,n}({\mathbf{X}},G)\}.\] If \(e=(1,\dots, 1)\), then \(\{G_{e,n}({\mathbf{X}})\}_{n\in \mathbb{N}}\) is independent of \({\mathbf{X}}\) (see [32]). In that case, we omit  \(e\) and \({\mathbf{X}}\), and write \(E_n\), \(G_n\), and  \(F_n\). This filtration coincides with the Zassenhaus filtration (see [34]) defined inductively by \[G_1\coloneq G, \quad \text{and} \quad G_{n}\coloneq G_{\lceil \frac{n}{p} \rceil}^p \prod_{i+j=n} \lbrack G_i, G_j\rbrack.\]

We endow \(R\) with the induced filtration given by \(R_{e,n}\coloneq F_{e,n}({\mathbf{X}})\cap R\). Let us observe that the filtration \(\{G_{e,n}({\mathbf{X}})\}_{n\in \mathbb{N}}\) coincides with the quotient filtration \(\{F_{e,n}({\mathbf{X}})\}_n\) by \(\{R_{e,n}\}_n\). We define \({\mathcal{J}}_e(R)\) the Lie ideal of \({\mathcal{L}}_e({\mathbf{X}})\) given by \({\rm Grad}(\varPsi_{F({\mathbf{X}})})\left(\bigoplus_{n\in \mathbb{N}} R_{e,n}/R_{e,n+1}\right)\), and we define: \[{\mathcal{L}}_e({\mathbf{X}}, G)\coloneq {\mathcal{L}}_e({\mathbf{X}})/{\mathcal{J}}_e(R)\coloneq \bigoplus_{n\in \mathbb{N}}{\mathcal{L}}_{e,n}({\mathbf{X}}, G).\] Note that \({\mathcal{L}}_{e,n}({\mathbf{X}}, G)\simeq G_{e,n}({\mathbf{X}})/G_{e,n+1}({\mathbf{X}})\).

For \(1\leq i \leq d\), we define \(n_{e,i}\coloneq \omega_e(\psi_{F({\mathbf{X}})}(l_i)-1)\), and \(\rho_{e,i}\) the image of \(\psi_{F({\mathbf{X}})}(l_i)-1\) in \({\mathcal{E}}_{e,n_{e,i}}({\mathbf{X}})\). We denote by \({\mathcal{I}}_e({\mathcal{P}})\) the two sided ideal generated by \(\{\rho_{e,i}\}_{i=1}^r\), and: \[{\mathcal{E}}_e({\mathbf{X}},{\mathcal{P}})\coloneq {\mathcal{E}}_e({\mathbf{X}})/{\mathcal{I}}_e({\mathcal{P}}), \quad \text{and} \quad {\mathcal{E}}_e({\mathbf{X}}, {\mathcal{P}},t)\coloneq {\mathcal{E}}_e({\mathbf{X}}, {\mathcal{P}})(t).\] Note that \(\rho_{e,i}\) can also be seen as an element of \({\mathcal{L}}_e({\mathbf{X}})\) through the injection (given by universal enveloping algebra theory) \({\mathcal{L}}_e({\mathbf{X}})\hookrightarrow {\mathcal{E}}_e({\mathbf{X}})\). Thus, we define \({\mathcal{J}}_e({\mathcal{P}})\) the restricted Lie-two sided ideal of \({\mathcal{L}}_e({\mathbf{X}})\) generated by \(\rho_{e,i}\), and \({\mathcal{L}}_e({\mathbf{X}},{\mathcal{P}})\coloneq {\mathcal{L}}_e({\mathbf{X}})/{\mathcal{J}}_e({\mathcal{P}})\).

1.3 Mild presentations↩︎

We continue to use the notation introduced in the previous section. In particular, \(G\) denotes a finitely presented pro-\(p\) group and \({\mathcal{P}}\) a presentation of \(G\). Little is known about the structure of \(G\) when \(\psi_{\mathcal{P}}\) and \(\varphi_{\mathcal{P}}\) fail to be isomorphisms. We therefore focus on situations in which they are isomorphisms. For this purpose, we use the following definition of mild groups (cf. [11]).

Let us recall that the notion of combinatorially free families provides an effective criterion to check mildness.

The next subsection provides concrete examples.

1.4 Right Angled Artin Algebras and group applications↩︎

Let \(\Gamma\coloneq ({\mathbf{X}}, {\mathbf{E}})\) be an (undirected) graph, and \(e\) some weights. For now, we fix a group \(G\), presented by \({\mathcal{P}}\) with set of generators \({\mathbf{X}}\) and relations \(\{l_{x_u,x_v}\}_{\{x_u,x_v\}\in {\mathbf{E}}}\).

If \(G\) is \(({\mathbf{X}},e)\)-RAAG for \(\Gamma\), then the initial form of the relations can now be determined as \(\rho_{e, x_u,x_v}=\mu(x_u,x_v)\lbrack X_u, X_v\rbrack=\mu(x_u,x_v)(X_uX_v-X_vX_u).\) We define \[{\mathcal{E}}_e({\mathbf{X}}, \Gamma) \coloneq {\mathcal{E}}_e({\mathbf{X}})/{\mathcal{I}}_e(\Gamma), \quad {\mathcal{L}}_e({\mathbf{X}}, \Gamma) \coloneq {\mathcal{L}}_e({\mathbf{X}})/{\mathcal{J}}_e(\Gamma)\quad \text{and} \quad {\mathcal{E}}_e({\mathbf{X}}, \Gamma,t) \coloneq {\mathcal{E}}_e({\mathbf{X}}, \Gamma)(t),\] where \({\mathcal{I}}_e(\Gamma)\) (resp.\({\mathcal{J}}_e(\Gamma)\)) is the ideal of \({\mathcal{E}}_e({\mathbf{X}})\) (resp.\({\mathcal{L}}_e({\mathbf{X}})\)) generated by \(\lbrack X_u, X_v\rbrack\) for \(\{x_u,x_v\}\in {\mathbf{E}}\). If \({\mathcal{P}}\) is the presentation of \(G\) associated with \({\mathbf{X}}\) and \(\{l_{x_u,x_v}\}_{\{x_u,x_v\}\in {\mathbf{E}}}\), then it is easy to observe that \({\mathcal{E}}_e({\mathbf{X}},\Gamma)\simeq {\mathcal{E}}_e({\mathbf{X}}, {\mathcal{P}})\) and \({\mathcal{L}}_e({\mathbf{X}}, \Gamma)\simeq {\mathcal{L}}_e({\mathbf{X}}, {\mathcal{P}})\).

We say that a graph \(\Gamma\coloneq ({\mathbf{X}},{\mathbf{E}})\) is bipartite if there is a nontrivial partition \({\mathbf{X}}\coloneq U \amalg V\), such that there are no edges between two vertices of \(U\), and no edges between two vertices of \(V\).

Before stating the next result, we introduce some definitions and notations. We define the algebra \({\mathcal{A}}(\Gamma)\) as the quadratic algebra over \(\mathbb{F}_p\) presented by generators \(\{X_1,\dots, X_d\}\) and relations:

•  \(X_iX_j\) when \(\{x_i,x_j\}\notin {\mathbf{E}}\),

•  \(X_uX_v+X_vX_u\) for \(x_u,x_v\) in \({\mathbf{X}}\).

Observe that \(\dim_{\mathbb{F}_p}{\mathcal{A}}_n(\Gamma)=c_n(\Gamma)\), where \(c_n(\Gamma)\) is the number of \(n\)-cliques of \(\Gamma\), i.e.complete subgraphs of \(\Gamma\) with \(n\) vertices. We also denote by \(H^\bullet(G)\) the (continuous) cohomology graded algebra of \(G\) with coefficients in \(\mathbb{F}_p\).

2 Koch-type presentations and linking numbers↩︎

We fix an odd prime \(p\). Throughout the paper, we assume that \(K\) is a number field, not containing \(\zeta_p\). We denote by \(r_K\) the \(\mathbb{Z}\)-rank of the unit group of \(\mathcal{O}_K\). If \((r_1,r_2)\) is the signature of \(K\), then \(r_K=r_1+r_2-1\) by the Dirichlet unit theorem. We also define \({\rm cl}(K)\) to be the class group of \(K\) and \(d_p{\rm cl}(K)\) to be its \(p\)-rank. This allows us to introduce \(d_K'\coloneq r_K+d_p{\rm cl}(K)\). We fix a set of primes \(T_K\) of \(K\) with Frobenii generating the \(p\)-part of the group \({\rm cl}(K)\). Note that \(|T_K|=d_p{\rm cl}(K)\).

Let \(S_1\), \(S_2\) be sets of primes of \(K\). Then we denote \[\begin{align} V_{S_1}^{S_2}(K)\coloneq \big\{a\in K^{\times }\mid a\in K_{\mathfrak q}^{\times p}\text{ for }{\mathfrak q}\in S_1\text{ and }p\mid \nu_\mathfrak{t}(a)\text{ for }\mathfrak{t}\not\in S_2\big\}/K^{\times p}, \end{align}\] with \(\nu_\mathfrak{t}\) the normalized valuation associated to \(\mathfrak{t}\). The dual of \(V_{S_1}^{S_2}(K)\) is denoted by \({{\textrm{\rurm{B}}}}_{S_1}^{S_2}(K)\) (see [37]). The set \({{\textrm{\rurm{B}}}}_{S_1}^{S_2}\) is important because it governs the Tate–Shafarevich group in Galois cohomology and is often computationally accessible (cf. proof of [theo-Liu], [38]). We note that if \(S_1\subseteq S_1'\) and \(S_2\subseteq S_2'\), then we have the following canonical surjections: \[\begin{align} {{\textrm{\rurm{B}}}}_{S_1}^{S_2}\twoheadrightarrow {{\textrm{\rurm{B}}}}_{S_1'}^{S_2},\qquad {{\textrm{\rurm{B}}}}_{S_1}^{S_2'}\twoheadrightarrow {{\textrm{\rurm{B}}}}_{S_1}^{S_2},\quad \text{and}\quad {{\textrm{\rurm{B}}}}_{S_1}^{S_2'}\twoheadrightarrow {{\textrm{\rurm{B}}}}_{S_1'}^{S_2}. \end{align}\]

By [37], if \(S\) and \(T\) are disjoint sets of primes, then the dimension of \({{\textrm{\rurm{B}}}}_S^{T}(K)\) determines the minimal number \(d(G_S^T)\) of generators of \(G_S^T\). More precisely, we have the following:

2.1 Koch sets↩︎

In general, knowledge of \({{\textrm{\rurm{B}}}}_S^{T}\) is not sufficient to determine the minimal number of relations of \(G_S^T\), let alone the concrete relations. In favorable situations, as considered by Liu in [20] or Salle in [21] one can determine the relations explicitly. We summarize their considerations in the form of Koch sets. More precisely, fix a minimal set of generators \(T_K\) of the \(p\)-part of \(\mathrm{cl}(K)\). We say that a finite set of tame primes \(S'\) of \(K\) is Koch if \(S'\cap T_K=\emptyset\) and \[\begin{align} {{\textrm{\rurm{B}}}}_{S'}^{T_K}(K)=1,\qquad \text{and}\qquad G_{K,S'}^{T_K}=1. \end{align}\] We say that a finite set \(S\) of tame primes of \(K\) admits a Koch set if \(S\cap T_K=\emptyset\) and there is \(S'\subseteq S\) such that \(S'\) is a Koch set.

For an integer \(n\), we define \(K_n\coloneq K(\zeta_{p^n})\). Before showing the existence of Koch sets, we fix some notations which we will use frequently. Given two disjoint sets of primes \(S\) and \(T\) we set \[\begin{align} \mathrm{Gov}^T_S(K)\coloneq K_1\Big(\sqrt[p]{V_S^{T}}\Big). \end{align}\] By Kummer duality it is not hard to see that \({\rm Gal}(\mathrm{Gov}_S^T(K)/K_1)\cong {{\textrm{\rurm{B}}}}_{S}^{T}\). If \(T=\emptyset\) (resp. \(S=\emptyset\)) we simply write \(\mathrm{Gov}_S(K)\) (resp. \(\mathrm{Gov}^T(K)\)). If \(K\) is clear from the context, we simply write \(\mathrm{Gov}_S^T\).

We briefly discuss statistics on Koch sets. Let \(T_K\) be a fixed set of primes of \(K\) such that their classes form a minimal generating set of \({\rm cl}(K)/p\). Under these assumptions, we obtain the following proposition.

2.2 Koch-type presentations and linking data↩︎

Fix a prime \(p\), and two integers \(d\) and \(d'\). Given two finite sets \({\mathbf{X}}=\{x_1,...,x_d\}\) and \({\mathbf{X}}'=\{x_1',...,x_{d'}'\}\). A linking tuple \(\lambda\) is a quintuple \(\lambda\coloneq ({\mathbf{X}}',{\mathbf{X}},\varepsilon,\mu,\omega)\) consisting of the following data:

• a function \(\varepsilon:{\mathbf{X}}'\times {\mathbf{X}}\to \{0,1\}\), to which we associate for each \(x'\in {\mathbf{X}}'\) a set \(\mathrm{Supp}_\varepsilon(x')\coloneq\{x\in {\mathbf{X}}\mid \varepsilon(x',x)=1\}\);

• a function \(\mu:D_\varepsilon\to \mathbb{F}_p\) where \[\begin{align} D_\varepsilon=\bigg\{(x,y)\in ({\mathbf{X}}\amalg {\mathbf{X}}')\times {\mathbf{X}}\left|{x\in{\mathbf{X}}\text{ and }y\neq x\text{ or }\atop x\in {\mathbf{X}}' \text{ and }y\in{\mathbf{X}}\setminus \mathrm{Supp}_\varepsilon(x)}\right.\bigg\}; \end{align}\]

• a sequence \(\omega\coloneq (\omega_z)_{z\in {\mathbf{X}}'\amalg {\mathbf{X}}}\) with \(0\neq \omega_z \in p\mathbb{Z}_p\) for \(z\in {\mathbf{X}}\amalg {\mathbf{X}}'\).

If \({\mathbf{X}}'=\emptyset\), we simply write \(({\mathbf{X}},\mu,\omega)\) instead of the full quintuple.

Now, let \(K\) be a number field not containing \(\zeta_p\), and we fix \(T_K\). Let \(S\) be a finite set of tame primes admitting a Koch set \(S'\). We construct a linking tuple \(\lambda_S^T\) in the following way. Let \({\mathbf{X}}'\coloneq S'\) and \({\mathbf{X}}\coloneq S\setminus S'\). Then by Lemma [lem:Unramified32exact32sequence] we have \[\begin{align} (K_S^T)^{\rm el,ab}\simeq \bigoplus_{x\in {\mathbf{X}}}\mathbb{F}_px. \end{align}\] For \(x'\in {\mathbf{X}}'\) fix a generator \(\tau_{x'}\) of the inertia group of \(x'\) in \((K_S^T)^{\rm el,ab}/K\). For \(x\in {\mathbf{X}}\) let \(\mu(x',x)\) be the \(x\)-coordinate of \(\tau_{x'}\). We also fix a Frobenius \(\varphi_{x'}\) in the maximal subextension of \((K_S^T)^{\rm el,ab}/K\) unramified at \(x'\) and set \(\mu(x',x)\) to be the \(x\) coordinate of \(\varphi_{x'}\) for \(x\) with \(\varepsilon(x',x)=0\). It is not hard to see that this value is indeed well defined.

Similarly, for \(x\in {\mathbf{X}}\) we let \(\varphi_x\) be a Frobenius in \((K_{S\setminus \{x\}}^T)^{\rm el,ab}/K\) and let for \(y\in S\setminus \{x\}\) be \(\mu(x,y)\) the \(y\)-coordinate of \(\varphi_x\). We set \(\omega_z\coloneq N({\mathfrak q}_z)-1\), where \({\mathfrak q}_{z}\) is the prime ideal corresponding to \(z\in {\mathbf{X}}\amalg {\mathbf{X}}'\).

These constitute \(\lambda_S^{T_K}=({\mathbf{X}}',{\mathbf{X}},\varepsilon,\mu,\omega)\). If we want to distinguish the data in the linking tuple from another one, we sometimes write \(\mu_S^{T_K}\) instead of \(\mu\), etc..

The following Theorem shows how \(\lambda_S^{T_K}\) and \(G_S^{T_K}\) are related:

2.3 Realization of linking data in the number field case↩︎

In [9] Labute asked in case \(K=\mathbb{Q}\) — in our notation — whether for every linking tuple \(\lambda=({\mathbf{X}},\mu,\omega)\), there exists a set of tame primes \(S\) such that one has \(\mu=\mu_S\) for \(\lambda_S=({\mathbf{X}}_S,\mu_S,\omega_S)\) under a suitable identification of \(S\) and \({\mathbf{X}}\).

The following theorem answers his question positively for arbitrary number fields not containing a primitive \(p\)-th root of unity. Throughout this subsection, we fix an arbitrary number field \(K\) satisfying \(\zeta_p \not\in K\) together with \(T_K\).

There is a more general form of the above theorem, for which we need the notation of extension of linking tuples. Let \(\lambda=({\mathbf{X}}',{\mathbf{X}},\varepsilon,\mu,\omega)\) and \(\widetilde{\lambda}=({\mathbf{X}}',\widetilde{{\mathbf{X}}},\widetilde{\epsilon},\widetilde{\mu},\widetilde{\omega})\) be linking tuples. We say that \(\widetilde{\lambda}\) extends \(\lambda\) if \({\mathbf{X}}\subseteq \smash{\widetilde{{\mathbf{X}}}}\) and the functions \(\varepsilon\), \(\smash{\widetilde{\varepsilon}}\) and \(\mu\), \(\smash{\widetilde{\mu}}\) are equal, wherever both of them are defined. Note that \(D_\varepsilon\subseteq D_{\smash{\widetilde{\varepsilon}}}\) under these assumptions.

Note that Theorem [thm:Realization32of32extended32linking32data] follows from Lemma [lem:Existence32of32Koch32Tuples] together with Theorem [extension]. Furthermore, Theorem [extension] follows from an inductive application of the following technical lemma, which is based on a method developed by Maire and Sankara in [22]:

3 Mildness of \(G_{K,S}^T\)↩︎

We recall that \(K\) is a number field not containing \(\zeta_p\). We fix \(T_K\) a minimal generating set of \({\rm cl}(K)/p\). We prove Theorem 1.

As a direct consequence, we infer Corollary 1.

4 Genus problems for \(G_{K,S}^T\) and graph theory↩︎

We follow the terminology of Leoni [29], which we adapt to our context. Let \(\Gamma\coloneq ({\mathbf{X}}, {\mathbf{E}})\) be an undirected graph. We define the genus of \(\Gamma\) as the set of pro-\(p\) groups: \[{\rm gen}(\Gamma)\coloneq \{G \text{ such that } {\mathcal{L}}(G)\simeq {\mathcal{L}}(\Gamma)\}.\] We note that the pro-\(p\) RAAG \(G_\Gamma\) (defined in Example [exem:32def32RAAG]) is always in \({\rm gen}(\Gamma)\). Thus the previous set is never empty. Furthermore, for a fixed \(\Gamma\), all pro-\(p\) groups in \({\rm gen}(\Gamma)\) share nice common properties. One of the most notable properties is cohomology. Precisely, as shown for instance in [23], every pro-\(p\) group \(G\) in \({\rm gen}(\Gamma)\) satisfies an isomorphism of graded algebras \(H^\bullet(G)\simeq {\mathcal{A}}(\Gamma)\).

We now return to the cohomological inverse problem stated in the introduction. We fix a number field \(K\) not containing \(\zeta_p\). For which graphs \(\Gamma\) do there exist a finite set of tame primes \(S\) and a finite disjoint set of primes \(T\) such that \(G_{K,S}^{T}\) is in \({\rm gen}(\Gamma)\)? If \(G_{K,S}^{T}\) answers the previous question positively, then we note, using the FAB property, that \(G_{K,S}^{T}\) is not isomorphic to \(G_\Gamma\).

Proposition [prop:rank32of32G95S94T] and Theorem [theo-Liu] already give some obstructions on the structure of \(\Gamma\). Yet, as noted in the introduction, we currently have no tools to handle the case where the cohomological dimension of \(G_{K,S}^{T}\) is finite but strictly greater than \(2\). This limitation naturally leads us to consider triangle-free graphs, a choice justified by Proposition [1-RAAG].

4.1 Pseudoforest graphs↩︎

This section exhibits a family of graphs relevant to the genus problem. We say that a graph \(\Gamma=({\mathbf{X}}, {\mathbf{E}})\) is a pseudoforest if each connected component has at most one cycle. If \(|{\mathbf{X}}|=|{\mathbf{E}}|\), then each connected component has precisely one cycle by Euler’s formula. We call those graphs strict pseudoforests. We first record the following lemma.

A graph \(\Gamma=({\mathbf{X}},{\mathbf{E}})\) is said to be inertial if it contains a strict pseudoforest \(({\mathbf{X}},{\mathbf{E}}')\) such that \({\mathbf{E}}'\subseteq {\mathbf{E}}\). In particular, for a number field \(K\), the graph \(\Gamma\) is called \(K\)-inertial if it is inertial with \(|{\mathbf{E}}|\geq d_K' + |{\mathbf{X}}|\).

4.2 The result for the genus problem in arithmetic↩︎

Let us now state the result of this section.

5 Numerical examples and computations↩︎

5.1 Examples Theorem 1↩︎

It is not hard to find numerical examples of \(2(1+d_K')\) additional tame primes, satisfying Theorem 1, in concrete situations. All the computations were made using the computer algebra system OSCAR [30]. The relevant source code can be found at [31]. Write \(S_0=\{q_1,q_2,...,q_d\}\). Then we are looking for tame primes \(q_{a_1}, q_{a_2}\), \(q_{1,1},\dots , q_{1,d_K'}\), \(q_{2,1},\dots ,q_{2,d_K'}\) such that the the linking numbers satisfy the relations from the proof of Theorem [thm:Mildness32for32arbitrary32fields]. In our search we were minimizing the size of \(q_{a_1}\) and chose the other primes accordingly. We give examples in the rational and quadratic real cases.

5.1.1 Rational case↩︎

Take \(p=3\). For \(S_0=\{7,13,19,37\}\) one finds that the set of tame primes \(S_1\coloneq \{10639,826093\}\) completes \(S_0\) to satisfy Corollary 1.

5.1.2 Real quadratic case↩︎

Fix \(p=3\). We give two examples:

(a) Consider \(K=\mathbb{Q}(\sqrt{3})\), we have \(d_K'=1\). We note that \(S'=\{(5)\}\) yields a Koch set for \(K\). Consider \(S_0\coloneq \{(7),(-\sqrt{3}+4)\}\). Following the proof of Theorem [thm:Mildness32for32arbitrary32fields] we are able to construct \[\begin{align} S_1\coloneq S'\amalg \{(1063),(593),(531\sqrt{3}+2224)\} \end{align}\] such that \(G_{K,\smash{\widetilde{S}}}\) with \(\widetilde{S}=S_0\cup S_1\) is mild.

(b) For \(K=\mathbb{Q}(\sqrt{79})\), we compute that \(d'_K=2\) and set \(T_K\coloneq \{(3,\sqrt{79}+1)\}\). We set \(S_0=\{(7,\sqrt{79}+4),(13,\sqrt{79}+1)\}\). Then \(S'=\{(7,\sqrt{79}+3),(11)\}\) is a Koch set disjoint from \(S_0\) and we find that for \[\begin{align} S_1\coloneq S'\amalg \big\{&(65\sqrt{79}+66),(17389,\sqrt{79}+4273),\\ &(-2919\sqrt{79}+26594),(86121199, \sqrt{79} + 5709782) \big \} \end{align}\] and \(\smash{\widetilde{S}}\coloneq S_0\cup S_1\) the group \(G_{K,\widetilde{S}}^{T_K}\) is mild.

5.2 Examples for \(1\)-RAAGs↩︎

We give two examples. One in the rational case, and the other in the real quadratic case. We take \(p=3\).

5.2.1 Rational case↩︎

Consider the case \(K=\mathbb{Q}\). Then \(d_K'=0\) and \(T_K=\emptyset\). Let us consider \(\Gamma\) the \(4\)-cycle graph with vertices \({\mathbf{X}}\coloneq \{x_1,x_2,x_3,x_4\}\) and edges \({\mathbf{E}}\coloneq \{\{x_1,x_2\}, \{x_2,x_3\}, \{x_3,x_4\}, \{x_4,x_1\}\}\). Then for \(S\coloneq \{7,13,181,5563\}\), the group \(G_S\) is a \(1\)-RAAG with respect to \(\Gamma\) and thus its cohomology is \({\mathcal{A}}(\Gamma)\).

5.2.2 Imaginary quadratic case↩︎

Let \(K=\mathbb{Q}(\sqrt{-31})\), then \(d_K'=1\). We consider the graph \(\Gamma\coloneq ({\mathbf{X}}, {\mathbf{E}})\) on six vertices and seven edges: \[{\mathbf{E}}\coloneq \{ \{x_1,x_2\}, \{x_2,x_3\}, \{x_3,x_4\}, \{x_4,x_5\}, \{x_5,x_6\}, \{x_6,x_1\}, \{x_1,x_4\}\}.\] The set \(T_K\coloneq \{(2,\tfrac{\sqrt{-31}+3}{2})\}\) generates \({\rm cl}(K)\cong \mathbb{Z}/3\) and \(S'=\{(7,\sqrt{-31}+2)\}\) is a Koch set. Furthermore, if we set \[\begin{align} S\coloneq S'\amalg\{&(7, \sqrt{-31} + 5), (13), (4129), (3613),\\& (257443, \sqrt{-31} + 245436), (1062643, \sqrt{-31} + 778479)\} \end{align}\] then \(G_{K,S}^{T_K}\) is a \(1\)-RAAG for \(\Gamma\) and hence \(H^\bullet(G_{K,S}^{T_K})\simeq {\mathcal{A}}(\Gamma)\).

5.3 Examples which are not \(({\mathbf{X}},e)\)-RAAG↩︎

Fix three finite sets \({\mathbf{X}}=\{x_1,\dots,x_d\}\), \({\mathbf{X}}'\coloneq \{x_1',\dots, x'_{d'}\}\) and \({\mathbf{X}}_0\coloneq \{z_1,\dots z_m\}\). Let \({\mathcal{L}}\coloneq{\mathcal{L}}({\mathbf{X}}).\) We introduce a family \(\rho \coloneq \{\rho_{x}, \rho_{x'}, \rho_z|\quad x\in {\mathbf{X}}, x'\in {\mathbf{X}}', z\in {\mathbf{X}}_0\}\) in \({\mathcal{L}}_2\). We denote by \({\mathcal{L}}(\rho)\) the quotient of \({\mathcal{L}}\) by the Lie-ideal generated by \(\rho\) and \({\mathcal{A}}(\rho)\) the quadratic dual of \({\mathcal{E}}(\rho)\). Here \({\mathcal{E}}(\rho)\) is the quotient of \({\mathcal{E}}\coloneq{\mathcal{E}}({\mathbf{X}})\) by the two-sided ideal generated by \(\rho\). For instance, we refer to [45] for definitions and references on quadratic duals.

Since \(p\) is odd, then \({\mathcal{L}}_2=\lbrack {\mathcal{L}}_1,{\mathcal{L}}_1\rbrack\). We assume that for every \(x_i \in {\mathbf{X}}\), there are coefficients \(a_{ij} \in \mathbb{F}_p\) such that \(\rho_{x_i} \coloneq \sum_{x_j\in {\mathbf{X}}, x_j\neq x_i} a_{ij}\lbrack X_i,X_j\rbrack\). For every \(x_k'\in {\mathbf{X}}'\), we assume there exists a tuple \((u_{k},v_{k})\in \Pi({\mathbf{X}})\) such that \(\rho_{x_k'}\coloneq \lambda_{x_k'}\lbrack \tau_{u_k}, \tau_{v_k}\rbrack\) with \(\lambda_{x_k'}\in \mathbb{F}_p^\ast\). This allow us to define a function \(\varepsilon_\rho \colon {\mathbf{X}}' \times {\mathbf{X}}\to \{0,1\}\) by \[\begin{align} \varepsilon_\rho(x_k',x)=\begin{cases} 1&\text{if }x=u_k,\\ 0&\text{otherwise.} \end{cases} \end{align}\] Thus, we have \(\mathrm{Supp}_{\varepsilon_\rho}(x_k')=\{u_k\}\), so \((x_k',v_k)\) is in \(D_{\varepsilon_\rho}\), and we have: \[D_{\varepsilon_\rho}\coloneq \Pi({\mathbf{X}})\cup \{(x_k', x_{k_i})|\quad x_k'\in {\mathbf{X}}, x_{k_i}\neq u_k\}.\] Thus, the family \(\rho\) induces a map \(\mu_\rho\colon D_{\varepsilon_\rho} \to \mathbb{F}_p\) defined by: \[\mu_{\rho}(x_i,x_j)\coloneq a_{ij} \quad \text{and} \quad \mu_{\rho}(x_k',v_k)\coloneq \lambda_{x_k'}, \quad \text{else } \mu_\rho(x_k',x)\coloneq 0,\] for \(x_i\in {\mathbf{X}}\), and \(x_k'\in {\mathbf{X}}'\). In particular, we have \(\rho_{x_i} \coloneq \sum_{x_j\neq x_i} \mu_{\rho}(x_i,x_j)\lbrack X_i,X_j\rbrack\).

References↩︎