Valuations of $p$-adic regulators of cyclic cubic fields

TL;DR

This paper computes $p$-adic regulators of cyclic cubic fields with large discriminants, analyzes their valuations, and proposes a probabilistic model and conjecture for their distribution.

Contribution

It introduces a computational approach to $p$-adic regulators of cyclic cubic fields and formulates a new conjecture based on observed statistical patterns.

Findings

Distribution of $p$-adic regulator valuations matches the random matrix model for most primes.

Empirical data supports the conjecture on the distribution of valuations.

Provides extensive computational data up to discriminant $10^{16}$.

Abstract

We compute the $p$-adic regulator of cyclic cubic extensions of $\mathbb Q$ with discriminant up to $10^{16}$ for $3<p<100$, and observe the distribution of the $p$-adic valuation of the regulators. We find that for almost all primes, the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model, a conjecture on the distribution of the valuations of $p$-adic regulators of cyclic cubic fields is stated.