The Manin–Mumford conjecture for Drinfeld modules

From papers

Let KK be a function field of transcendence degree 11 over Fp\mathbb{F}_p, and let ϕ:AK{τ}\phi:A\rightarrow K\{\tau\} be a Drinfeld module. Let n1n\ge 1 and let XX be an irreducible KalgK^{\operatorname{alg}}-subvariety of Gan\mathbb{G}_a^n. The torsion points for ϕ\phi are points of Gan(Kalg)\mathbb{G}_a^n(K^{\operatorname{alg}}) whose coordinates are torsion points for the diagonal action. Manin–Mumford conjecture for Drinfeld modules. If the set of torsion points PGan(Kalg)P\in\mathbb{G}_a^n(K^{\operatorname{alg}}) for ϕ\phi is dense in XX, then there exist a torsion point γϕtorn\overline{\gamma}\in\phi_{\operatorname{tor}}^n and a KalgK^{\operatorname{alg}}-subvariety YGanY\subset\mathbb{G}_a^n, invariant under a nonzero endomorphism of ϕ\phi, such that

X=γ+Y.X=\overline{\gamma}+Y.

This is presented as a weak form of the Bogomolov conjecture. The examples in the supplied context indicate that, in finite characteristic, invariance under an endomorphism is the sharp conclusion, while the source suggests a stronger formulation may hold in generic characteristic; no resolution of this stated conjecture is supplied.

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Sources & referencesView supporting material

Primary source

Dragos Ghioca, “Points of small height on varieties defined over a function field”, arXiv:math/0505001 (2005).

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