math.NT — Number Theory

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

Contributed by Shawn Krisman

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.