A Random Group with Local Data Realizing Heuristics for Number Field Counting

Abstract

We define a group with local data over a number field $K$ as a group $G$ together with homomorphisms from decomposition groups ${\rm Gal}(\overline{K}_p/K_p)\to G$. Such groups resemble Galois groups, just without global information. Motivated by the use of random groups in the study of class group statistics, we use the tools given by Sawin-Wood to construct a random group with local data over $K$ as a model for the absolute Galois group ${\rm Gal}(\overline{K}/K)$ for which representatives of Frobenius are distributed Haar randomly as suggested by Chebotarev density. We utilize Law of Large Numbers results for categories proven by the author to show that this is a random group version of the Malle-Bhargava principle. In particular, it satisfies number field counting conjectures such as Malle's Conjecture under certain notions of probabilistic convergence including convergence in expectation, convergence in probability, and almost sure convergence. These results produce new heuristic justifications for number field counting conjectures, and begin bridging the theoretical gap between heuristics for number field counting and class group statistics.