Arakelov theory [SEP] named after
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.
Arakelov geometry studies a scheme "X" over the ring of integers Z, by putting Hermitian metrics on holomorphic vector bundles over "X"(C), the complex points of "X". This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety.
Hodge–Arakelov theory
In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and "p"-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by .
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"-dimensional space of functions on the "d"-torsion