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

Shawn Krisman

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

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

Contributed by Shawn Krisman

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

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.

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

References

Armand Brumer, On the units of algebraic number fields , Mathematika 14 (1967), 121--124.
Samit Dasgupta, Mahesh Kakde, and Kevin Ventullo, On the G ross-- S tark conjecture , Ann. of Math. (2) 188 (2018), no. 3, 833--870.
Fabio Ferri and Henri Johnston, Applications of representation theory and of explicit units to L eopoldt's conjecture , Res. Number Theory 12 (2026), no. 2, Paper No. 32.
Georges Gras, Class field theory: From theory to practice , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
— , The $p$ -adic K ummer-- L eopoldt constant: normalized $p$ -adic regulator , Int. J. Number Theory 14 (2018), no. 2, 329--337.
— , On $p$ -rationality of number fields. A pplications -- PARI/GP programs , Publ. Math. Besan ç on Alg è bre Th é orie Nr. (2019), no. 2, 29--51.
Tommy Hofmann and Yinan Zhang, Valuations of $p$ -adic regulators of cyclic cubic fields , J. Number Theory 169 (2016), 86--102.
Jean-Fran ç ois Jaulent and Thong Nguyen Quang Do, Corps $p$ -rationnels, corps $p$ -r é guliers, et ramification restreinte , J. Th é or. Nombres Bordeaux 5 (1993), no. 2, 343--363.
Henri Johnston and Andreas Nickel, On the $p$ -adic S tark conjecture at $s=1$ and applications , J. Lond. Math. Soc. (2) 101 (2020), no. 3, 1320--1354, With an appendix by T. Hofmann, H. Johnston and A. Nickel.
Helmut Koch, Galois theory of $p$ -extensions , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.

Show all 16 referencesShow fewer references

Shawn Krisman, A modular obstruction to the $p$ -adic L eopoldt conjecture for the C arlitz module , 2026, Preprint.
Heinrich-Wolfgang Leopoldt, Zur A rithmetik in abelschen Z ahlk ö rpern , J. Reine Angew. Math. 209 (1962), 54--71.
Hang Li and Derong Qiu, On $p$ -rationality of cubic and quartic number fields , 2023, arXiv:2304.10157.arXiv
Abbas Movahhedi and Thong Nguyen Quang Do, Sur l'arithm é tique des corps de nombres $p$ -rationnels , S é minaire de T h é orie des N ombres, P aris 1987--88, Progr. Math., vol. 81, Birkh ä user Boston, 1990.
J ü rgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields , 2 ed., Grundlehren der mathematischen Wissenschaften, vol. 323, Springer-Verlag, Berlin, 2008.
The PARI Group , PARI/GP version 2.17.3 , 2025, http://pari.math.u-bordeaux.fr/.link

Version history

v1Submitted by Shawn KrismanInitial depositSupersededOct 24, 2026
v2Submitted by Shawn KrismanSubmitted as a new versionCurrentOct 26, 2026