Heuristics for p-class towers of imaginary quadratic fields

By Nigel Boston, M. Bush, F. Hajir. 2011

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Cohen and Lenstra have given a heuristic which, for a fixed odd prime p, leads to many interesting predictions about the distribution of p-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field K, the Galois group of the p-class tower of K, i.e. $${G}_K:=\mathrm {Gal}(K_\infty /K)$$GK:=Gal(K∞/K) where $$K_\infty $$K∞ is the maximal unramified p-extension of K. By class field theory, the maximal abelian quotient of $${G}_K$$GK is isomorphic to the p-class group of K. For integers $$c\ge 1$$c≥1, we give a heuristic of Cohen-Lenstra type for the maximal p-class c quotient of $${G}_K$$GK and thereby give a conjectural formula for how frequently a given p-group of p-class c occurs in this manner. In particular, we predict that every finite Schur $$\sigma $$σ-group occurs as $$G_K$$GK for infinitely many fields K. We present numerical data in support of these conjectures.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/79b32118992cf52af5fc13e3fb8ec5190f37e476