mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Hodge–Arakelov theory of

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

s is an analogue of classical and

p-adic Hodge theory In mathematics, ''p''-adic Hodge theory is a theory that provides a way to classify and study ''p''-adic Galois representations of characteristic 0 local fields with residual characteristic ''p'' (such as Q''p''). The theory has its beginnings i ...

for elliptic curves carried out in the framework of

Arakelov theory In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. Background The main motivation behind Arakelov geometry is tha ...

. It was introduced by . It bears the name of two mathematicians,

Suren Arakelov Suren Yurievich Arakelov (, ) (born October 16, 1947 in Kharkiv) is a Soviet mathematician of Armenian descent known for developing Arakelov theory. Biography From 1965 onwards Arakelov attended the Mathematics department of Moscow State Univers ...

and

W. V. D. Hodge Sir William Vallance Douglas Hodge (; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer. His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area no ...

. The main comparison in his theory remains unpublished as of 2019. Mochizuki's main comparison theorem in Hodge–Arakelov theory states (roughly) that the space of

polynomial function In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative int ...

s of degree less than ''d'' on the universal extension of a smooth elliptic curve in characteristic 0 is naturally

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

(via restriction) to the ''d''²-dimensional space of functions on the ''d''- torsion points. It is called a 'comparison theorem' as it is an analogue for Arakelov theory of comparison theorems in cohomology relating

de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...

singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

of complex varieties or

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...

of ''p''-adic varieties. In and he pointed out that arithmetic

Kodaira–Spencer map In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a Map (mathematics), map associated to a Deformation theory, deformation of a Scheme (mathematics), scheme or complex manifold ''X'', taking a tangen ...

and

Gauss–Manin connection In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s ...

may give some important hints for

Vojta's conjecture In mathematics, Vojta's conjecture is a conjecture introduced by about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribu ...

ABC conjecture ABC are the first three letters of the Latin script. ABC or abc may also refer to: Arts, entertainment and media Broadcasting * Aliw Broadcasting Corporation, Philippine broadcast company * American Broadcasting Company, a commercial American ...

and so on; in 2012, he published his Inter-universal Teichmuller theory, in which he didn't use Hodge-Arakelov theory but used the theory of

frobenioid In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by . The word "Frobenioid" is a portmanteau of ...

s, anabelioids and mono-anabelian geometry.

References

* * * {{DEFAULTSORT:Hodge-Arakelov theory Number theory Algebraic geometry Abc conjecture

See also

References