Abstract
Let be a complete discrete valuation ring with the finite residue field . Given a monic polynomial whose reduction modulo gives an irreducible polynomial , we initiate an investigation of the distribution of , where is randomly chosen with respect to the Haar probability measure on the additive group of R-matrices. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. We use some concrete combinatorial connections between and to translate our problems about a Haar-random matrix in into problems about a random matrix in with respect to the uniform distribution. Our results over are about the distribution of the -part of a random matrix with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., -module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of with a probability inversely proportional to . We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems.
Citation
Gilyoung Cheong. Yifeng Huang. "Cohen–Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields." Illinois J. Math. 65 (2) 385 - 415, June 2021. https://doi.org/10.1215/00192082-8939615
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