Vol. 12, No. 2, 2019

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Numerical secondary terms in a Cohen–Lenstra conjecture on real quadratic fields

Codie Lewis and Cassandra Williams

Vol. 12 (2019), No. 2, 221–233
DOI: 10.2140/involve.2019.12.221

In 1984, Cohen and Lenstra made a number of conjectures regarding the class groups of quadratic fields. In particular, they predicted the proportion of real quadratic fields with class number divisible by an odd prime. We numerically investigate the difference between reality and these predictions. Using 4 million data points, we perform a curve fitting of the difference with a monomial term and demonstrate that there is reason to believe the term can be effectively approximated within the scope of our data set for odd primes less than 30. We use cross-validation to show that including our monomial term as a secondary term to the original conjecture reduces the overall error.

Cohen–Lenstra, real quadratic field, secondary term
Mathematical Subject Classification 2010
Primary: 11R29
Secondary: 11R11, 11Y35
Received: 15 June 2017
Revised: 8 March 2018
Accepted: 11 June 2018
Published: 8 October 2018

Communicated by Yves-François Pétermann
Codie Lewis
Department of Mathematics
Colorado State University
Fort Collins, CO
United States
Cassandra Williams
Department of Mathematics and Statistics
James Madison University
Harrisonburg, VA
United States