Schweizer's torsion conjecture for rank-two Drinfeld modules

Let AA be the coefficient ring, let FF be the base field, and let ϕ\phi be a rank-22 Drinfeld AA-module over FF. Write

(ϕF)torA/mA/n,mn.(^\phi F)_\mathrm{tor}\cong A/\mathfrak{m}\oplus A/\mathfrak{n},\qquad \mathfrak{m}\mid\mathfrak{n}.

Schweizer's Drinfeld-module torsion conjecture.

degm+degn2.\deg\mathfrak{m}+\deg\mathfrak{n}\leq 2.

Equivalently, if p\mathfrak{p} is a prime of AA of degree at least 33, then Y1(p)Y_1(\mathfrak{p}) has no FF-rational points. The source later says that the conjecture remains largely open as of 2024.

Sources & referencesView supporting material

Primary source

Cécile Armana, Sheng-Yang Kevin Ho and Mihran Papikian, “Ogg's conjectures over function fields”, arXiv:2410.05502 (2024).

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