21/08/28 11:42:18.13 j6A6Uinw.net
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Anabelian geometry is intensively used in Mochizuki’s IUT = arithmetic deformation theory and its applications to some of the abc inequalities, and the Szpiro and Vojta conjectures, [49], [50]. It is interesting to observe
that similarly to the Neukirch explicit CFT and the Vostokov symbol in explicit formulas for the Hilbert pairing,
IUT involves several indeterminacies at its crucial stage of multi-radial representation. IUT uses generalised
Kummer theory and the computation of the local Brauer group, it does not use anything else from CFT. It
works with values of certain nonarchimedean functions (etale theta functions) at torsion points, in this respect
it is nearer to SCFT; on the other hand, it works over any number field and in this respect it is nearer to GCFT.21
Informally speaking, IUT deals with Galois groups as tangent bundles, see the beginning of sect. 2.6 and
4.3 (ii) of [50]. To a certain degree, global class field theory does kind of the same with abelian Galois groups:
abelian Galois groups over a global field correspond to idele classes, while adeles are dual to generalised
differential forms.
21 See also Remark 2.3.3 of IUT-IV paper [49] and sect. 4.2 of [50]
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