Abstract
In this paper, we study the anabelian geometry of hyperbolic polycurves of dimension $2$ over sub-$p$-adic fields. In $1$-dimensional case, Mochizuki proved the Hom version of the Grothendieck conjecture for hyperbolic curves over sub-$p$-adic fields and the pro-$p$ version of this conjecture. In $2$-dimensional case, a naive analogue of this conjecture does not hold for hyperbolic polycurves over general sub-$p$-adic fields.Moreover, the Isom version of the pro-$p$ Grothendieck conjecture does not hold in general. We explain these two phenomena and prove the Hom version of the Grothendieck conjecture for hyperbolic polycurves of dimension $2$ under the assumption that the Grothendieck section conjecture holds for some hyperbolic curves.
Acknowledgments
The author thanks Yuichiro Hoshi for various useful comments, and especially for the following: (i) informing me of the arguments used in Theorem 3.4; (ii) explaining to me various results about the Grothendieck section conjecture.
This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
Citation
Ippei Nagamachi. "Topics in the Grothendieck conjecture for hyperbolic polycurves of dimension $2$." Osaka J. Math. 61 (1) 91 - 105, January 2024.
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