{"id":"hexagon/2610.00532","pdf":"https://hexagonmath.org/pdf/2610.00532v1","url":"https://hexagonmath.org/abs/2610.00532v1","tags":["msc:11R27","msc:11R16","msc:11R34","msc:11R37","msc:11R29","Galois cohomology","Leopoldt conjecture","Poitou-Tate duality","S3 extensions","cup products"],"score":0,"title":"Forced $3$-adic anomalies of $S_3$ cubic fields","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Shawn Krisman"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Fable","Claude Opus"],"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}