A modular obstruction to the $P$-adic Leopoldt conjecture for the Carlitz module

Shawn Krisman

A modular obstruction to the $P$ -adic Leopoldt conjecture for the Carlitz module

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

Contributed by Shawn Krisman ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

Let

L

be a finite extension of

K=\mathbb{F}_q(\theta)

and

P

a monic prime of

\mathbb{F}_q[\theta]

. Anglès asked whether the

P

-adic Leopoldt defect of Taelman's module of units of the Carlitz module over

L

always vanishes, and Lucas conjectured this, together with the equivalence "

L_P(E/\mathcal{O}_L)\neq 0

iff

\exp_E

is injective", for Anderson modules. We show that both statements fail. For every odd prime

p

, the extension of

\mathbb{F}_p(\theta)

generated by a root of

z^p-(-\theta)^{p-2}z-(-\theta)^{p-1}

is real, has Galois group

\mathrm{AGL}(1,p)

, and has positive

P

-adic Leopoldt defect and vanishing

P

-adic zeta value at

s=1

for every

P

. The obstruction comes from modular representation theory at the infinite place. We attach to every separable

L/K

an integer

\delta_{\mathrm{gen}}(L)

, depending only on the Galois group of the Galois closure, its decomposition group at

\infty

and the character by which this group acts on the Carlitz period, and prove that the

P

-adic defect is at least

\delta_{\mathrm{gen}}(L)

for all

P

. Using Bosser's

P

-adic Baker-Brumer theorem, we prove that the defect vanishes for every

P

for real abelian extensions and for abelian extensions with cyclic Sylow

p

-subgroup. We also show that a

v

-adic analogue of Papanikolas's algebraic independence theorem for Carlitz logarithms would imply that the defect equals

\delta_{\mathrm{gen}}(L)

in general.

Provenance statement

This was done with a Fable Claude Code session for the research and Opus for the paper.

References

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Version history

v1Submitted by Shawn KrismanInitial depositCurrentOct 24, 2026