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The mean number of 3‐torsion elements in the class groups and ideal groups of quadratic orders
Author(s) -
Bhargava Manjul,
Varma Ila
Publication year - 2016
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdv062
Subject(s) - mathematics , torsion (gastropod) , quadratic equation , pure mathematics , ideal class group , ideal (ethics) , combinatorics , algebraic number field , geometry , medicine , surgery , philosophy , epistemology
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 ; andI 3 ( O ) , the group of ideals of order 3 . We determine the mean values of both| Cl 3( O ) |and| I 3( O ) | , as O 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 fields 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.