math.NT — Number Theory

Forced $3$-adic anomalies of $S_3$ cubic fields

Contributed by Shawn Krisman

Let $k$ be a totally real cubic field and $p$ a prime unramified in $k$. The $p$-adic regulator of $k$ is a $2\times2$ determinant of $p$-adic logarithms of units and is always divisible by $p^2$; call $k$ anomalous at $p$ if it is divisible by $p^3$. Among all 592421 totally real cubic fields of discriminant up to $10^7$, the proportion of anomalous fields matches a random-matrix prediction at every prime $5\le p\le 31$, but at $p=3$ it is $0.381$ instead of the predicted $0.407$. We explain this. For cubic fields of fundamental discriminant $D$ prime to 3, we show that the 3-class group and the ray class group modulo 9 of the quadratic field $\mathbb{Q}(\sqrt{D})$ can force every cubic field of discriminant $D$ to be anomalous at 3. When the 3-class group of $\mathbb{Q}(\sqrt{D})$ is cyclic, we decide exactly when $k$ is anomalous, in terms of one invariant for which we give an explicit formula by cubic residue and Hilbert symbols. The phenomenon comes from the failure of semisimplicity of $\mathbb{F}_3[S_3]$ and has no analogue for $p\ge 5$. The results fix one or two 3-adic digits of the regulator and say nothing about Leopoldt's conjecture itself.