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Random pro-\(p\) groups, braid groups, and random tame Galois groups. (English) Zbl 1239.11126

The author introduces a heuristic prediction for the distribution of the isomorphism class of the Galois group of the maximal pro-\(p\) extension of \(\mathbb Q\) unramified outside a random set of primes. This is guided by reasoning similar to that governing the Cohen-Lenstra conjectures.
Let \(S\) be a set of primes in \(\mathbb Q\), \(G_S(p)\) the Galois group of the maximal \(p\)-extension unramified away from \(p\) (including infinity if \(p=2\)). The aim of the article is to present and justify a heuristic answer to the question: When \(S\) is an random set of primes, what is the probability that \(G_S(p)\) is isomorphic to some specified finite \(p\)-group \(\Gamma\)?
More precisely, let \(S:=(\ell_1,\dots,\ell_g)\) be a \(g\)-tuple of primes \(\equiv 1\mod p\), let \(Z_i\) the closure of \(\ell_i^{\mathbb Z}\) in \(\mathbb Z_p^*\) and \(W_i\) the group \(\mathbb Z_p/(\ell_i-1)\mathbb Z_p\). The group \(G_S^{ab}\) is isomorphic to \(W:=\bigoplus_{i=1}^g W_i\), finite abelian \(p\)-group of rank \(g\). The type of \(S\) is the sequence of subgroups \((Z_1,\dots,Z_g)\). We write \(W(Z)\) for the finite \(p\)-group attached to a type \(Z=(Z_1,\dots, Z_g)\) by taking the sum of the corresponding \(W_i\). Let \(Z=(Z_1,\dots,Z_g)\) be a type and \(\Gamma\) be a finite \(p\)-group such that \(\Gamma^{ab}\cong W(Z)\) and \(h_1(\Gamma,{\mathbb F}_p)=h_2(\Gamma,{\mathbb F}_p)=g\) where \(h_i(\Gamma,{\mathbb F}_p)\) denotes the dimension of \(H_i(\Gamma,{\mathbb F}_p)\), (so \(\Gamma\) is balanced). Let \(P(Z,\Gamma,X)\) be the proportion of \(g\)-tuples of primes \(S(\ell_1,\dots,\ell_g)\) with type \(Z\) in \([X,...,2X]^g\) such that \(G_S(p)\cong \Gamma\). The behavior of \(P(Z,\Gamma, X)\) as \(X\) grows can be thought of as the probability that a random \(g\)-tuple of primes of type \(Z\) has \(\Gamma\) as its maximal unramified Galois group.
Let \(A_Z(\Gamma)\) be the number of pairs \(((c_1,\dots,c_g),l)\) where \((c_1,\dots,c_g)\) is a \(g\)-tuple of conjugacy classes in \(\Gamma\) and \(l\) is an involution in \(\Gamma\) (trivial automatically when \(p\) is odd) such that:
\(c_i^z=c_i\) for all \(z\in \mathbb Z_i\);
The elements \(\pi(c_1),\dots,\pi(c_g)\) generate \(\Gamma^{ab}\) where \(\pi:\Gamma\rightarrow\Gamma^{ab}\) is the natural projection;
The map \(W(Z)\rightarrow \Gamma^{ab}\) sending \((w_1,\dots w_g)\) to \(\sum_i w_i\pi(c_i)\) is an isomorphism;
If \(p=2\) and \(l_i\) is the unique nontrivial involution in the cyclic subgroup of \(\Gamma^{ab}\) generated by \(\pi(c_i)\), we have \(\sum_{i=1}^g l_i=\pi(i)\).
The author states the following:
Heuristic: \(\lim_{X\rightarrow \infty} P(Z,\Gamma, X)\) exists and is equal to \(A_Z(\Gamma)/|\operatorname{Aut}(\Gamma)|\).
The author gives:
1)
some justifications for the random \(p\)-groups with inertia data corresponding to \(G_S(p)\),
2)
some justifications for the random pro-\(p\) groups and random pro-\(p\) braid groups with an analogy with functions fields,
3)
for \(p=2\), the previous ones completed with experimental tests with MAGMA software,
4)
some numerical evidences for the groups with abelianization \(\mathbb Z/p\mathbb Z\times \mathbb Z/p\mathbb Z\), (\(p\) odd), \(\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z\), \(\mathbb Z/2\mathbb Z\times \mathbb Z/4\mathbb Z\) and \(\mathbb Z/4\mathbb Z\times \mathbb Z/4\mathbb Z\).

MSC:

11R32 Galois theory
20E18 Limits, profinite groups
20D15 Finite nilpotent groups, \(p\)-groups

Citations:

Zbl 1188.11058
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References:

[1] J. D. Achter, Results of Cohen-Lenstra type for quadratic function fields. In Computa- tional arithmetic geometry , Contemp. Math. 463,Amer. Math. Soc., Providence, RI 2008, 1-7. · Zbl 1166.11018
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