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

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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.

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Notes

  1. Technically speaking, the pullback \(j_p^*\) appearing here may be thought of as a pullback of locally analytic functions on associated p-adic analytic spaces.

  2. Technically speaking, while the map \({\text {loc}}_v\) appearing in the diagram is a morphism of \({\mathbb {Q}_p}\)-schemes, the vertical maps j, \(j_v\) are just maps of sets into the sets of \({\mathbb {Q}_p}\)-points of the varieties below.

  3. To avoid misunderstanding, we remind the reader that \(\mathcal {E}\) refers to the compact curve, so that \(\mathcal {E}(\mathbb {Z})=E(\mathbb {Q})\), where E is the generic fiber of \(\mathcal {E}\). That is, what we write as \(\mathcal {E}(\mathbb {Z})\) is what is usually called the rational points of E, while our \(\mathcal {X}(\mathbb {Z})\) is sometimes confusingly referred to as the integral points of E.

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Acknowledgements

M.K. is grateful to John Coates, Henri Darmon, Kazuya Kato, Florian Pop, and Andrew Wiles for a continuous stream of discussions on the topic of this paper. He is also grateful to Shinichi Mochizuki whose question prompted a precise formulation of the conjecture, and to Yuichiro Hoshi for a kind and detailed reply to a question about a pro-p analogue. We would like to thank the referee for many helpful comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA, 02215, USA

    Jennifer S. Balakrishnan

  2. Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK

    Minhyong Kim

  3. Department of Mathematics, Ewha Women’s University, 52 Ewha-yeo-dae-gil, Seoul, 120-750, South Korea

    Minhyong Kim

  4. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, Israel

    Ishai Dan-Cohen

  5. Institut für Reine Mathematik, Universität Ulm, Helmholtzstrasse 18, 89081, Ulm, Germany

    Stefan Wewers

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  1. Jennifer S. Balakrishnan
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  2. Ishai Dan-Cohen
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  3. Minhyong Kim
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  4. Stefan Wewers
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Correspondence to Ishai Dan-Cohen.

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Communicated by A. Venkatesh.

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Balakrishnan, J.S., Dan-Cohen, I., Kim, M. et al. A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves. Math. Ann. 372, 369–428 (2018). https://doi.org/10.1007/s00208-018-1684-x

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