Coleman's unramified torsion conjecture for curves with good reduction

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Let p≥5p\geq 5 be a prime number, and suppose that K/QpK/{\mathbb Q}_p is an unramified finite extension. Let X/KX/K be a curve of genus g≥2g\geq 2, embedded in its Jacobian via a KK-rational Albanese map. Suppose furthermore that XX has good reduction over KK. A torsion point P∈X(K‾)P\in X(\overline{K}) is unramified if its field of definition is unramified over KK. Coleman's conjecture. Every torsion point P∈X(K‾)P\in X(\overline{K}) is unramified. This is an open problem concerning the Galois action on torsion points lying on curves embedded in their Jacobians; the stated source presents it as an intriguing open problem and attributes it to R. Coleman.

References

Primary source

Matthew Baker and Kenneth A. Ribet, “Galois theory and torsion points on curves”, arXiv:math/0212133 (2002).

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