The Semisimplicity Conjecture for A-Motives

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Akio Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G_p(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G_p(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V_p(M). The second states that the connected component of G_p(M) is reductive if M is semisimple and has a separable endomorphism algebra.Comment: 47 page

A refined version of Grothendieck's birational anabelian conjecture for curves over finite fields

In this paper we prove a refined version of Uchida's theorem on isomorphisms between absolute Galois groups of global fields in positive characteristics, where one "ignores" the information provided by a "small" set of primes.Comment: Final version, to appear in Advances in Mathematics (2017

A refined version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields

In this paper we prove a refined version of a theorem by Tamagawa and Mochizuki on isomorphisms between (tame) arithmetic fundamental groups of hyperbolic curves over finite fields, where one "ignores" the information provided by a "small" set of primes.Comment: 55 pages, to appear in Journal of Algebraic Geometry. arXiv admin note: text overlap with arXiv:0801.145