A geometric approach to the Cohen-Lenstra heuristics

A geometric approach to the Cohen-Lenstra heuristics

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Aaron Landesman, Stanford University




The Cohen-Lenstra heuristics in number theory describe the average number of $n$-torsion elements in class groups of quadratic fields. We present a geometric approach  to these heuristics, which has the following consequence: under the correspondence between quadratic forms and elements of class groups of quadratic fields, a quadratic form $q$ corresponds to an $n$-torsion element if and only if there exists a degree $n$ polynomial whose resultant with $q$ is $\pm 1$.This follows from a general structure theorem describing $\mu_n$ torsors over degree $2$ covers, which is based on a classical construction in algebraic geometry.