Strong arithmetic Shafarevich conjecture for semistable curves

Let K(C)K(C) be a field as above, and let ρ\rho be an ideal of the field FnrF^{\mathrm{nr}}. Strong arithmetic Shafarevich conjecture. There exists a finite extension FF of KK and a semistable model CC' of F(C)F(C) such that the ring ResρRes_\rho is a direct sum of finitely many fields for every such ideal ρ\rho. This is proposed as a strong arithmetic analogue of Shafarevich's conjecture; the source does not state what is known or whether it has been resolved.

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Primary source

Fedor Bogomolov and Ludmil Katzarkov, “Complex projective surfaces and infinite groups”, arXiv:alg-geom/9703002 (1997).

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