Coleman's conjecture on the ramification of torsion points of curves

Let CC be a curve of genus g2g \geq 2 embedded in its Jacobian JJ, all defined over a number field KK. Let JtorsJ_{\mathrm{tors}} be the torsion subgroup of JJ and let Ctors:=C(Kˉ)JtorsC_{\mathrm{tors}}:= C(\bar{K}) \cap J_{\mathrm{tors}}. Let p\mathfrak{p} be a prime ideal of OK\mathcal{O}_K above a rational prime pp such that p5p \geq 5, K/QK/\mathbb{Q} is unramified at p\mathfrak{p}, and CC has good reduction at p\mathfrak{p}. Coleman's conjecture. Then the extension K(Ctors)/KK(C_{\mathrm{tors}})/K is unramified above p\mathfrak{p}. The conjecture concerns quantitative refinements of Raynaud's theorem for curves. Its resolution is not established by the supplied text, so its status remains open here.

Sources & referencesView supporting material

Primary source

Aurélien Galateau and César Martínez, “A bound for the torsion on subvarieties of abelian varieties”, arXiv:1710.04577 (2017).

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