In mathematics, 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 ...

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 ...

for elliptic curves carried out in the framework of Arakelov theory. It was introduced by . It bears the name of two mathematicians,

Suren Arakelov Suren Yurievich Arakelov (russian: Суре́н Ю́рьевич Араке́лов, arm, Սուրե՛ն Յուրիի՛ Առաքելո՛վ) (born October 16, 1947 in Kharkiv) is a Soviet mathematician of Armenian descent known for developing Arakel ...

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 now c ...

. The main comparison in his theory remains unpublished as of 2019.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 functions of degree less than ''d'' on the universal extension of a smooth elliptic curve in characteristic 0 is naturally isomorphic (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 adap ...

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 viewe ...

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 conject ...

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 associated to a deformation of a scheme or complex manifold ''X'', taking a tangent space of a point of the deformation space to the fir ...

and Gauss–Manin connection may give some important hints for Vojta's conjecture,

ABC conjecture The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b'' ...

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 frobenioids, anabelioids and mono-anabelian geometry.

References

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

See also

References