Grothendieck Conjecture for Hyperbolic Curves over Finitely Generated Fields of Positive Characteristic via Compatibility of Cyclotomes

Abstract

Let p be a prime number. In the present paper, from the viewpoint of the compatibility/rigidity of group-theoretic cyclotomes, we revisit the anabelian Grothendieck Conjecture for hyperbolic curves over finitely generated fields of characteristic p established by A. Tamagawa, J. Stix, and S. Mochizuki. Especially, we give an alternative proof of the Grothendieck Conjecture for nonisotrivial hyperbolic curves over finitely generated fields of characteristic p obtained by J. Stix. In fact, by applying relatively recent results in anabelian geometry for hyperbolic curves over finite fields developed by M. Saïdi and A. Tamagawa, we discuss the J. Stix's result in a certain generalized situation, i.e., the geometrically pro-Σ setting, where Σ denotes the complement of a finite set of prime numbers that contains p in the set of all prime numbers. Moreover, by combining with a theorem in birational anabelian geometry obtained by F. Pop, we prove an absolute version of the geometrically pro-Σ Grothendieck Conjecture for nonisotrivial hyperbolic curves over the perfections of finitely generated fields of characteristic p. On the other hand, in the present paper, we also establish certain isotriviality criteria for hyperbolic curves with respect to both l-adic Galois representations and pro-l outer Galois representations, where l is a prime number ≠ p. These isotriviality criteria may be applied to give an alternative proof of the J. Stix's result.