Mono-anabelian Reconstruction of Number Fields (On the examination and further development of inter-universal Teichmuller theory)

Abstract

The Neukirch - Uchida theorem asserts that every outer isomorphism between the absolute Galois groups of number fields arises from a uniquely determined isomorphism between the given number fields. In particular, the isomorphism class of a number field is completely determined by the isomorphism class of the absolute Galois group of the number field. On the other hand, neither the Neukirch-Uchida theorem nor the proof of this theorem yields an "explicit reconstruction of the given number field". In other words, the Neukirch-Uchida theorem only yields a bi-anabelian reconstruction of the given number field. In the present paper, we discuss a mono-anabelian reconstruction of the given number field. In particular, we give afunctorial "group-theoretic" algorithm for reconstructing, from the absolute Galois group of a number field, the algebraic closure of the given number field [equipped with its natural Galois action] that gave rise to the given absolute Galois group. One important step of our reconstruction algorithm consists of the construction of a global cyclotome [i.e., a cyclotome constructed from a global Galois group] and a local - global cyclotomic synchronization isomorphism [i.e., a suitable isomorphism between a global cyclotome and a local cyclotome]. We also verify a certain compatibility between our reconstruction algorithm and the reconstruction algorithm given by S. Mochizuki concerning the étale fundamental groups of hyperbolic orbicurves of strictly Belyi type over number fields. Finally, we discuss acertain global mono - anabelian log - Frobenius compatibility property satisfied by the reconstruction algorithm obtained in the present paper.