Geometric Version of the Grothendieck Conjecture for Universal Curves over Hurwitz Stacks: a research announcement (Algebraic Number Theory and Related Topics 2017)

Abstract

Hurwitz stacks are algebraic stacks that parametrize simple coverings. In this paper, we introduce a certain geometric version of the Grothendieck Conjecture for universal curves over Hurwitz stacks. This result generalizes a similar result obtained by Hoshi and Mochizuki in the case of universal curves over moduli stacks of pointed smooth curves. After introducing these results, we give a sketch of the proof of the above version of the Grothendieck Conjecture in the hyperelliptic case.