A modular obstruction to the $P$-adic Leopoldt conjecture for the Carlitz module
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…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2610.00525
curl -X POST /api/v1/records/hexagon/2610.00525/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00525/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'