Poonen's uniform boundedness conjecture for torsion of Drinfeld modules

Fix qq, r1r\geq 1, and d1d\geq 1. Let LL range over extensions of FF of degree at most dd, and let ϕ\phi range over rank-rr Drinfeld AA-modules over LL. Poonen's Drinfeld-module uniform boundedness conjecture. The cardinalities #(ϕL)tors\#(^\phi L)_\mathrm{tors} are uniformly bounded. For rank r=2r=2, equivalently, there is a constant C>0C>0 such that whenever degnC\deg\mathfrak{n}\geq C and [L:F]d[L:F]\leq d, the curve Y1(n)Y_1(\mathfrak{n}) has no LL-rational points. This is the Drinfeld-module analogue of uniform boundedness for torsion of elliptic curves and abelian varieties. The supplied text gives no resolution.

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Primary source

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

Additional references

3 papers in this index state this conjecture (1995–2024). The statement above is taken from the most recent of them; the others are arXiv:1210.3059, arXiv:math/9507217.

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