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Forced 33-adic anomalies of S3S_3 cubic fields

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Shawn Krisman

Version 2 / Oct 26, 2026 / CC BY 4.0

Abstract

Let kk be a totally real cubic field and pp a prime unramified in kk. The pp-adic regulator of kk is a 2×22\times2 determinant of pp-adic logarithms of units and is always divisible by p2p^2; call kk anomalous at pp if it is divisible by p3p^3. Among all 592421 totally real cubic fields of discriminant up to 10710^7, the proportion of anomalous fields matches a random-matrix prediction at every prime 5≤p≤315\le p\le 31, but at p=3p=3 it is 0.3810.381 instead of the predicted 0.4070.407. We explain this. For cubic fields of fundamental discriminant DD prime to 3, we show that the 3-class group and the ray class group modulo 9 of the quadratic field Q(D)\mathbb{Q}(\sqrt{D}) can force every cubic field of discriminant DD to be anomalous at 3. When the 3-class group of Q(D)\mathbb{Q}(\sqrt{D}) is cyclic, we decide exactly when kk 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 F3[S3]\mathbb{F}_3[S_3] and has no analogue for p≥5p\ge 5. The results fix one or two 3-adic digits of the regulator and say nothing about Leopoldt's conjecture itself.

Provenance statement

The core of the research was done with Claude Code Fable, the paper was written by Opus

Tools used

Anthropic
Claude FableVersion 5.1
Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1 · 11 pagesSubmitted by Shawn KrismanInitial depositSupersededOct 24, 2026
  2. v2 · 13 pagesSubmitted by Shawn KrismanSubmitted as a new versionCurrentOct 26, 2026