Compositio Mathematica

On the Mean 3-Rank of Quadratic Fields

a1 Université Paris-Sud, Département de Mathématiques (bât.42S), F-91405 Orsay, France e-mail:

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belabas k   [Google Scholar] 


The Cohen–Lenstra–Martinet heuristics give precise predictions about the class groups of a ’random‘ number field. The 3-rank of quadratic fields is one of the few instances where these have been proven. We prove that, in this case, the rate of convergence is at least sub-exponential. In addition, we show that the defect appearing in Scholz‘s mirror theorem is equidistributed with respect to a twisted Cohen–Lenstra density.

Key Words: class groups; heuristics; 3-rank; quadratic fields..