Nonabelian Cohen-Lenstra Heuristics over Function Fields

Abstract

Boston, Bush, and Hajir have developed heuristics, extending the Cohen-Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-p extensions of imaginary quadratic number fields for p an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of F_q(t), the Galois groups of the maximal unramified pro-p extensions, as q goes to infinity, have the moments predicted by the Boston, Bush, and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

minor corrections made, to appear in Compositio

References in corpus (1)

• Random matrices, the Cohen-Lenstra heuristics, and roots of unity

Cited by in corpus (6)

• Nonabelian Cohen-Lenstra Moments
• The Distribution of p-Torsion in Degree p Cyclic Fields
• Higher genus theory
• Statistics of $K$-groups modulo $p$ for the ring of integers of a varying quadratic number field
• Monogenicity and 2-torsion in the class group of number fields of odd degree
• Presentations of Galois groups of maximal extensions with restricted ramification