Good reduction of hyperbolic polycurves and their fundamental groups : A survey (Algebraic Number Theory and Related Topics 2018)

Algebraic Number Theory and Related Topics 2018. November 26-30, 2018. edited by Takao Yamazaki and Shuji Yamamoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The goal of this manuscript is to provide a survey of good reduction criteria for hyperbolic polycurves. In particular, we give outlines of the proofs of the main theorems of the papers [19] and [20], which are details of the talk “Criteria for good reduction of hyperbolic polycurves” given at “Algebraic Number Theory and Related Topics 2018”. Also, this paper contains a proof of a specialization theorem of pro-L fundamental groups

Criteria for good reduction of hyperbolic polycurves

We give good reduction criteria for hyperbolic polycurves, i.e., successive extensions of families of curves, under some assumptions. These criteria are higher dimensional versions of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa

TOPICS IN THE GROTHENDIECK CONJECTURE FOR HYPERBOLIC POLYCURVES OF DIMENSION 2

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