Poonen's uniform boundedness conjecture for torsion of Drinfeld modules

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Fix qq, r≥1r\geq 1, and d≥1d\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 deg⁡n≥C\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.

References

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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