1. Heuristics for p-class towers of imaginary quadratic fields
- Author
-
Nigel Boston, Michael R. Bush, and Farshid Hajir
- Subjects
Discrete mathematics ,Group (mathematics) ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Galois group ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Tower (mathematics) ,Combinatorics ,Class field theory ,Quadratic field ,0101 mathematics ,Abelian group ,Quotient ,Mathematics - Abstract
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. $${G}_K:=\mathrm {Gal}(K_\infty /K)$$ where $$K_\infty $$ is the maximal unramified p-extension of K. By class field theory, the maximal abelian quotient of $${G}_K$$ is isomorphic to the p-class group of K. For integers $$c\ge 1$$ , we give a heuristic of Cohen-Lenstra type for the maximal p-class c quotient of $${G}_K$$ 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 $$\sigma $$ -group occurs as $$G_K$$ for infinitely many fields K. We present numerical data in support of these conjectures.
- Published
- 2016