{"id":"hexagon/2610.00525","pdf":"https://hexagonmath.org/pdf/2610.00525v1","url":"https://hexagonmath.org/abs/2610.00525v1","tags":["math.RT","msc:11G09","msc:11R58","msc:11M38","msc:11J93","msc:11R27","Carlitz logarithms","Carlitz module","Drinfeld modules","Goss zeta function"],"score":0,"title":"A modular obstruction to the $P$-adic Leopoldt conjecture for the Carlitz module","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Shawn Krisman"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}