Anabelian geometry
Encyclopedia
Anabelian geometry is a proposed theory in mathematics
, describing the way the algebraic fundamental group G of an algebraic variety
V, or some related geometric object, determines how V can be mapped into another geometric object W, under the assumption that G is very far from being an abelian group
, in a sense to be made more precise. The word anabelian (an alpha privative an- before abelian) was introduced in Esquisse d'un Programme
, an influential manuscript of Alexander Grothendieck
, circulated in the 1980s.
While the work of Grothendieck was for many years unpublished, and unavailable through the traditional formal scholarly channels, the formulation and predictions of the proposed theory received much attention, and some alterations, at the hands of a number of mathematicians. Those who have researched in this area have obtained some expected and related results, and in the 21st century the beginnings of such a theory started to be available.
A concrete example is the case of curves, which may be affine as well as projective. Suppose given a hyperbolic curve C, i.e. the complement of n points in a projective algebraic curve
of genus g, taken to be smooth and irreducible, defined over a field K that is finitely generated (over its prime field), such that
Grothendieck conjectured that the algebraic fundamental group G of C, a profinite group, determines C itself (i.e. the isomorphism class of G determines that of C). This was proved by Shinichi Mochizuki. An example is for the case of g = 0 (the projective line
) and n = 4, when the isomorphism class of C is determined by the cross-ratio
in K of the four points removed (almost, there being an order to the four points in a cross-ratio, but not in the points removed). There are also results for the case of K a local field
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, describing the way the algebraic fundamental group G of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V, or some related geometric object, determines how V can be mapped into another geometric object W, under the assumption that G is very far from being an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, in a sense to be made more precise. The word anabelian (an alpha privative an- before abelian) was introduced in Esquisse d'un Programme
Esquisse d'un Programme
"Esquisse d'un Programme" is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck...
, an influential manuscript of Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
, circulated in the 1980s.
While the work of Grothendieck was for many years unpublished, and unavailable through the traditional formal scholarly channels, the formulation and predictions of the proposed theory received much attention, and some alterations, at the hands of a number of mathematicians. Those who have researched in this area have obtained some expected and related results, and in the 21st century the beginnings of such a theory started to be available.
Formulation of a conjecture of Grothendieck on curves
The "anabelian question" has been formulated asA concrete example is the case of curves, which may be affine as well as projective. Suppose given a hyperbolic curve C, i.e. the complement of n points in a projective algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
of genus g, taken to be smooth and irreducible, defined over a field K that is finitely generated (over its prime field), such that
- 2 – 2g – n < 0.
Grothendieck conjectured that the algebraic fundamental group G of C, a profinite group, determines C itself (i.e. the isomorphism class of G determines that of C). This was proved by Shinichi Mochizuki. An example is for the case of g = 0 (the projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
) and n = 4, when the isomorphism class of C is determined by the cross-ratio
Cross-ratio
In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
in K of the four points removed (almost, there being an order to the four points in a cross-ratio, but not in the points removed). There are also results for the case of K a local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
.
External links
- Heidelberg Lectures on Fundamental Groups, section 5.