Effective convergence of coranks of random R\'edei matrices

TL;DR

This paper provides effective estimates for the distribution of coranks in random Rédéi matrices, connecting stochastic process approximations with the Cohen–Lenstra heuristics, and extends results to related matrix families relevant to quadratic fields.

Contribution

It introduces the concept of c-transitioning stochastic processes and proves an effective ergodic theorem, advancing the understanding of Rédéi matrix distributions in number theory.

Findings

Effective bounds on the l^1-distance to Cohen–Lenstra predictions.

Extension of results to matrix families related to quadratic forms.

Application of stochastic process approximation to number theory problems.

Abstract

We give effective estimates for the $l^1$-distance between the corank distribution of $r \times r$ R\'edei matrices and the measure predicted by the Cohen--Lenstra heuristics. To this end we pinpoint a class of stochastic processes, which we call $c$-transitioning. These stochastic processes are well approximated by Markov processes, and we give an effective ergodic theorem for such processes. With this tool we make effective a theorem of Gerth \cite{Gerth} that initiated the study of the Cohen--Lenstra heuristics for $p = 2$. Gerth's work triggered a series of developments that has recently culminated in the breakthrough of Smith \cite{Smith}. The present work will be used in upcoming work of the authors on further applications of Smith's ideas to the arithmetic of quadratic fields. To this end we extend our main result to several other families of matrix spaces that occur in the study of integral points on the equation $x^2 - dy^2 = l$ as $d$ varies.