##
**On the distribution of class groups of number fields.**
*(English)*
Zbl 1297.11139

Let \( K_1/K_0\) a finite extension, \(L/K_0\) its Galois closure with Galois group \(G = \mathrm{Gal}(L/K_0)\). Then \(G\) acts on the different embeddings of \(K_1\) into \(L\) by the transitive permutation representation on its subgroup \(\mathrm{Gal}(L/K_1)\). The corresponding permutation character \(\chi\) contains the trivial character \(1_G\) exactly once. Let \(\chi_1 := \chi - 1_G\). Let \(\mathbb Q[G]\) denote the rational group ring of the Galois group \(G\). The author assumes that:

\(\chi_1\) is the character of an irreducible \(\mathbb Q[G]\)-module.

Any absolutely irreducible constituent \(\phi\) of \(\chi_1\) is the character of a representation of \(G\) over the field of values of \(\phi\).

Denote by \(E_L\) the group of units of the ring of integers of \(L\). Then the action of \(G\) makes \(E_L\otimes_{\mathbb Z} \mathbb Q\) into a \(\mathbb Q[G]\)-module, whose character is denoted by \(\chi_E\). Let \(u :=\langle \chi_E, \phi\rangle\) the scalar product of the character \(\chi_E\) with an absolutely irreducible constituent \(\phi\) of \(\chi_1\).

Let’s denote by \(\mathcal K(\Sigma)\) a situation, where \(\Sigma = (G,K_0, \sigma\)), the set of number fields \(K/K_0\) with signature \(\sigma\) and Galois group of the Galois closure permutation isomorphic to \(G\).

The author proposes the following

Conjecture : Assume that p does not divide the permutation degree of \(G\) and that \(K_0\) contains the \(p\)th but not the \(p^2\)th roots of unity. Then a given finite \(p\)-group \(H\) of \(p\)-rank \(r\) occurs as Sylow \(p\)-subgroup of a relative class group \(\mathrm{Cl}(K/K_0)\) for \(K \in \mathcal K(\Sigma)\) with probability

\[ c\cdot \frac{\prod_{i=u+1}^{u+r}(p^i-1)}{p^{r(u+1)}}\cdot \frac{1}{|H|^u|\mathrm{Aut}(H)|} \text{\;where\;} c=\frac{1}{\prod_{i=u+1}^\infty (1+p^{-i})}. \] The author presents numerical data for the distribution of \(p\)-parts of class groups for the following situations \(\Sigma\) and primes \(p\):

\(\Sigma = (C_2,\mathbb Q(\sqrt{-3}),\text{\;complex}), p=3\).

\(\Sigma = (C_2,\mathbb Q(\mu_5),\text{\;complex}), p=5\).

\(\Sigma = (\mathfrak S_3,\mathbb Q,\text{\;totally real}), p=2\).

\(\Sigma = (C_3,\mathbb Q,\text{\;totally real}), p=2\).

\(\Sigma = (C_3,\mathbb Q(\sqrt{-3}),\text{\;complex}), p=2\).

\(\Sigma = (C_3,\mathbb Q(\sqrt{5}),\text{\;totally real}), p=2\).

\(\Sigma = (C_3,\mathbb Q(\sqrt{-1}),\text{\;complex}), p=2\).

\(\Sigma = (D_5,\mathbb Q,\text{\;complex}), p=2\).

\(\Sigma = (D_5,\mathbb Q,\text{\;real}), p=2\).

Furthermore, the predicted formula agrees with results on class groups of function fields in positive characteristic for which the base field contains appropriate roots of unity. See for instance the works of H. Cohen and J. Martinet [J. Reine Angew. Math. 404, 39–76 (1990; Zbl 0699.12016)] and G. Malle [J. Number Theory 128, No. 10, 2823–2835 (2008; Zbl 1225.11143)].

\(\chi_1\) is the character of an irreducible \(\mathbb Q[G]\)-module.

Any absolutely irreducible constituent \(\phi\) of \(\chi_1\) is the character of a representation of \(G\) over the field of values of \(\phi\).

Denote by \(E_L\) the group of units of the ring of integers of \(L\). Then the action of \(G\) makes \(E_L\otimes_{\mathbb Z} \mathbb Q\) into a \(\mathbb Q[G]\)-module, whose character is denoted by \(\chi_E\). Let \(u :=\langle \chi_E, \phi\rangle\) the scalar product of the character \(\chi_E\) with an absolutely irreducible constituent \(\phi\) of \(\chi_1\).

Let’s denote by \(\mathcal K(\Sigma)\) a situation, where \(\Sigma = (G,K_0, \sigma\)), the set of number fields \(K/K_0\) with signature \(\sigma\) and Galois group of the Galois closure permutation isomorphic to \(G\).

The author proposes the following

Conjecture : Assume that p does not divide the permutation degree of \(G\) and that \(K_0\) contains the \(p\)th but not the \(p^2\)th roots of unity. Then a given finite \(p\)-group \(H\) of \(p\)-rank \(r\) occurs as Sylow \(p\)-subgroup of a relative class group \(\mathrm{Cl}(K/K_0)\) for \(K \in \mathcal K(\Sigma)\) with probability

\[ c\cdot \frac{\prod_{i=u+1}^{u+r}(p^i-1)}{p^{r(u+1)}}\cdot \frac{1}{|H|^u|\mathrm{Aut}(H)|} \text{\;where\;} c=\frac{1}{\prod_{i=u+1}^\infty (1+p^{-i})}. \] The author presents numerical data for the distribution of \(p\)-parts of class groups for the following situations \(\Sigma\) and primes \(p\):

\(\Sigma = (C_2,\mathbb Q(\sqrt{-3}),\text{\;complex}), p=3\).

\(\Sigma = (C_2,\mathbb Q(\mu_5),\text{\;complex}), p=5\).

\(\Sigma = (\mathfrak S_3,\mathbb Q,\text{\;totally real}), p=2\).

\(\Sigma = (C_3,\mathbb Q,\text{\;totally real}), p=2\).

\(\Sigma = (C_3,\mathbb Q(\sqrt{-3}),\text{\;complex}), p=2\).

\(\Sigma = (C_3,\mathbb Q(\sqrt{5}),\text{\;totally real}), p=2\).

\(\Sigma = (C_3,\mathbb Q(\sqrt{-1}),\text{\;complex}), p=2\).

\(\Sigma = (D_5,\mathbb Q,\text{\;complex}), p=2\).

\(\Sigma = (D_5,\mathbb Q,\text{\;real}), p=2\).

Furthermore, the predicted formula agrees with results on class groups of function fields in positive characteristic for which the base field contains appropriate roots of unity. See for instance the works of H. Cohen and J. Martinet [J. Reine Angew. Math. 404, 39–76 (1990; Zbl 0699.12016)] and G. Malle [J. Number Theory 128, No. 10, 2823–2835 (2008; Zbl 1225.11143)].

Reviewer: Roland Quême (Brax)

### MSC:

11R29 | Class numbers, class groups, discriminants |

11R16 | Cubic and quartic extensions |

11R21 | Other number fields |

11R58 | Arithmetic theory of algebraic function fields |

11Y40 | Algebraic number theory computations |

### References:

[1] | DOI: 10.1016/j.jpaa.2005.04.003 · Zbl 1134.11042 |

[2] | Achter J., Computational Arithmetic Geometry pp 1– (2008) |

[3] | Andrews G. E., Encyclopedia of Mathematics and Its Applications 2 (1976) |

[4] | DOI: 10.1090/S0025-5718-1987-0866103-4 |

[5] | Cohen H., J. Reine Angew. Math. 404 pp 39– (1990) |

[6] | DOI: 10.1515/crll.2002.071 · Zbl 1004.11063 |

[7] | DOI: 10.1006/jabr.2000.8455 · Zbl 0980.20037 |

[8] | DOI: 10.1016/j.jnt.2008.01.002 · Zbl 1225.11143 |

[9] | DOI: 10.1090/S0025-5718-00-01291-6 · Zbl 0985.11068 |

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