Taelman's conjecture for finite uniformizable t-modules

Let K=Fq(θ)K={\mathbb F}_q(\theta), let FF be a finite extension of KK, let OF{\cal O}_F be an integral closure of AA in FF, and let F=FKKF_{\infty}=F\otimes_K K_{\infty}. Let Υ\Upsilon be a finite uniformizable t\mathbf{t}-module over OF{\cal O}_F. Set

WΥ(F):=Lie(Υ)(F)/(Υ(θ)θId)Lie(Υ)(F)W_{\Upsilon}(F_{\infty}):={\mathrm{Lie}}(\Upsilon)(F_{\infty})/(\partial_{\Upsilon}(\theta)-\theta I_d){\mathrm{Lie}}(\Upsilon)(F_{\infty})

and let w:Lie(Υ)(F)WΥ(F)w:{\mathrm{Lie}}(\Upsilon)(F_{\infty})\to W_{\Upsilon}(F_{\infty}) be the projection. Write r:=dimKWΥ(F)r:=\dim_{K_{\infty}}W_{\Upsilon}(F_{\infty}). Taelman's conjecture. There exist an element aA\{0}a\in A\backslash\{0\} and a sub-AA-module ZLie(Υ)(F)Z\subset {\mathrm{Lie}}(\Upsilon)(F_{\infty}) of rank rr such that

expΥ(Z)Lie(Υ)(OF)\exp_{\Upsilon}(Z)\subset {\mathrm{Lie}}(\Upsilon)({\cal O}_F)

and

Arw(Z)=aL(Υ/OF)ArWΥ(OF).{\bigwedge}_A^r w(Z)=a\cdot L(\Upsilon/{\cal O}_F)\cdot {\bigwedge}_A^r W_{\Upsilon}({\cal O}_F).

This conjecture predicts an arithmetic description of a suitable lattice in the Lie algebra of a finite uniformizable t\mathbf{t}-module, relating its exterior power to the associated LL-value. The supplied text does not state whether the conjecture has been proved or disproved.

Sources & referencesView supporting material

Primary source

Dawid E. Kędzierski and Piotr Krasoń, “Homological Methods in the Generalization of Drinfeld Modules”, arXiv:2512.07607 (2026).

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