TY - JOUR

T1 - The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders

AU - Bhargava, Manjul

AU - Varma, Ila

N1 - Funding Information:
The first author was supported by the Packard and Simons Foundations and NSF Grant DMS-1001828. The second author was supported by a National Defense Science & Engineering Fellowship and an NSF Graduate Research Fellowship.
Publisher Copyright:
© 2015 London Mathematical Society.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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.

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U2 - 10.1112/plms/pdv062

DO - 10.1112/plms/pdv062

M3 - Article

AN - SCOPUS:84965170345

SN - 0024-6115

VL - 112

SP - 235

EP - 266

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 2

ER -