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. The main motivation behind Arakelov geometry is the fact there is a correspondence between prime ideals and finite places , but there also exists a place at infinity , given by the Archimedean valuation, which doesn't have a corresponding prime ideal. Arakelov geometry gives a technique for compactifying into a complete space which has a prime lying at infinity. Arakelov's original construction studies one such theory, where a definition of divisors is constructor for a scheme of relative dimension 1 over such that it extends to a Riemann surface for every valuation at infinity. In addition, he equips these Riemann surfaces with Hermitian metrics on holomorphic vector bundles over X(C), the complex points of . This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety. Note that other techniques exist for constructing a complete space extending , which is the basis of F1 geometry. Let be a field, its ring of integers, and a genus curve over with a non-singular model , called an arithmetic surface. Also, we let be an inclusion of fields (which is supposed to represent a place at infinity). Also, we will let be the associated Riemann surface from the base change to . Using this data, we can define a c-divisor as a formal linear combination where is an irreducible closed subset of of codimension 1, , and , and the sum represents the sum over every real embedding of and over one embedding for each pair of complex embeddings . The set of c-divisors forms a group . defined an intersection theory on the arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields, in the case of number fields.