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

Manjul Bhargava, Ila Varma

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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: Cl3(O), the group of ideal classes of order 3; and I3(O), the group of ideals of order 3. We determine the mean values of both |Cl3(O)| and |I3(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 |Cl3(O)|-|I3(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 |Cl3(O)|-1/3|I3(O)| is equal to 1, independent of the family.

Original languageEnglish (US)
Pages (from-to)235-266
Number of pages32
JournalProceedings of the London Mathematical Society
Volume112
Issue number2
DOIs
StatePublished - Feb 1 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Fingerprint Dive into the research topics of 'The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders'. Together they form a unique fingerprint.

Cite this