anabelian geometry

English

Noun

anabelian geometry (countable and uncountable, plural anabelian geometries)

1. (mathematics, algebraic geometry, arithmetic geometry) A theory which describes the way in which the algebraic fundamental group G of an algebraic variety (or some related geometric object) V determines how V can be mapped into another geometric object W, under the assumption that G is very far from being abelian (commutative).
   • 1999, Shinichi Mochizuki, Foundations of p-adic Teichmüller Theory, American Mathematical Society, page 63:

     We shall say more about the noncommutative nature of anabelian geometries after we state the main theorem (Theorem 1.17 below) on these geometries.

   • 2000, Florian Pop, “Alterations and Birational Anabelian Geometry”, in H. Hauser, J. Lipman, F. Oort, A. Quirós, editors, Resolution of Singularities, Birkhäuser, page 519:

     The basic idea of Grothendieck's anabelian geometry is that under certain "anabelian hypotheses" the étale fundamental group of a scheme contains all the geometric and arithmetic information about the scheme in discussion, that is to say, the scheme is functorially encoded in its étale fundamental group.
     Among these we shall discuss applications to the inverse Galois problem, Belyi's theorem on covers of the projective line minus three points and some advanced results on ‘anabelian geometry’ of curves.

   • 2012, John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page vii:

     Therefore, the time seems right to encourage a much broader understanding of the arithmetic issues surrounding anabelian geometry and its ramifications.
     While the overall importance of the theorems of anabelian geometry appears to be widely acknowledged, there is as yet not much specific knowledge within the arithmetic geometry community of its coherent body of concepts and philosophy, and of the new technology that yields actual results.

Translations

theory in algebraic geometry

• German: anabelsche Geometrie f
• Italian: please add this translation if you can