Abstract

We present a conjecture on the distribution of the valuations of $p$-adic regulators of cyclic extensions of $\mathbb{Q}$ of odd prime degree. This is based on the observation of computational data of $p$-adic regulators of the 5 521 222 cyclic quintic and 329 708 cyclic septic extensions of $\mathbb{Q}$ for $2<p<100$ with discriminant up to $5\times 10^{31}$ and $10^{42}$ respectively, and noting that the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions.

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Mathematical Subject Classification 2010

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