Conditionally on sparse 'strange' newforms, the largest prime order of rational torsion on elliptic curves over degree-d fields is at most 3d+1 for large even d and o(d) for odd d.
Low Degree Places on the Modular Curve X1(N)
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This short note gives questions and examples of points on X1(N) defined over number fields whose degrees are lower than the gonality.
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Prime order torsion on elliptic curves over number fields. Part I: Asymptotics
Conditionally on sparse 'strange' newforms, the largest prime order of rational torsion on elliptic curves over degree-d fields is at most 3d+1 for large even d and o(d) for odd d.