Forced $3$-adic anomalies of $S_3$ cubic fields
For a totally real cubic field $k$ in which a prime $p$ is unramified, the $p$-adic regulator $R_p(k)$ is a $2\times 2$ determinant of $p$-adic logarithms of units, and $v_p(R_p(k))\ge 2$ always. Call $k$ anomalous at $p$ if $v_p(R_p(k))\ge 3$. In the complete list of the 592421 totally real cubic fields of discriminant at most $10^7$, the anomaly rate at every prime $5\le p\le 31$ agrees with the Haar random-matrix value $1/p+1/p^2-1/p^3$, whereas at $p=3$ it is $0.381$ instead of $11/27\approx 0.407$. We explain the $p=3$ behaviour for fields of fundamental discriminant $D$ with $3\nmid…
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