# The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders

## Author(s): Bhargava, Manjul; Varma, Ila

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Abstract: | We determine the mean number of 3-torsion elements in the class groups of quadratic orders, where the quadratic orders are ordered by their absolute discriminants. Moreover, for a quadratic order O we distinguish between the two groups: Cl 3 (O), the group of ideal classes of order 3; and I 3 (O), the group of ideals of order 3. We determine the mean values of both |Cl 3 (O)| and |I 3 (O)|,asO ranges over any family of orders deﬁned by ﬁnitely many (or in suitable cases, even inﬁnitely many) local conditions. As a consequence, we prove the surprising fact that the mean value of the diﬀerence |Cl 3 (O)|− |I 3 (O)| is equal to 1, regardless of whether one averages over the maximal orders in complex quadratic ﬁelds or over all orders in such ﬁ elds or, indeed, over any family of complex quadratic orders deﬁned by local conditions. For any family of real quadratic orders deﬁned by local conditions, we prove similarly that the mean value of the diﬀerence |Cl 3 (O)|− 1 3 |I 3 (O)| is equal to 1, independent of the family. |

Publication Date: | Feb-2016 |

Electronic Publication Date: | Feb-2016 |

Citation: | Bhargava, Manjul, Varma, Ila. (2016). The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 112 (235 - 266). doi:10.1112/plms/pdv062 |

DOI: | doi:10.1112/plms/pdv062 |

ISSN: | 0024-6115 |

EISSN: | 1460-244X |

Pages: | 235 - 266 |

Type of Material: | Journal Article |

Journal/Proceeding Title: | PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY |

Version: | Author's manuscript |

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