A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves

Abstract

Let X denote a hyperbolic curve over $$\mathbb {Q}$$ and let p denote a prime of good reduction. The third author’s approach to integral points, introduced in Kim (Invent Math 161:629–656, 2005; Publ Res Inst Math Sci 45:89–133, 2009), endows $$X({\mathbb {Z}_p})$$ with a nested sequence of subsets $$X({\mathbb {Z}_p})_n$$ which contain $$X(\mathbb {Z})$$ . These sets have been computed in a range of special cases (Balakrishnan et al., J Am Math Soc 24:281–291, 2011; Dan-Cohen and Wewers, Proc Lond Math Soc 110:133–171, 2015; Dan-Cohen and Wewers, Int Math Res Not IMRN 17:5291–5354, 2016; Kim, J Am Math Soc 23:725–747, 2010); there is good reason to believe them to be practically computable in general. In 2012, the third author announced the conjecture that for n sufficiently large, $$X(\mathbb {Z}) = X({\mathbb {Z}_p})_n$$ . This conjecture may be seen as a sort of compromise between the abelian confines of the BSD conjecture and the profinite world of the Grothendieck section conjecture. After stating the conjecture and explaining its relationship to these other conjectures, we explore a range of special cases in which the new conjecture can be verified.