nLab Arakelov geometry

Motivation

The aim of Arakelov geometry is to extend intersection theory to the case of algebraic curves over $Spec(\mathbb{Z})$, hence in arithmetic geometry.

Arakelov complemented the algebraic geometry at finite primes with a holomorphic piece at a place at infinity. Then using complex analytic geometry and Green's functions he defined the intersection numbers using the complementary piece at infinity.

Definitions

… e.g. (Durov 07)

Properties

Arithmetic Riemann-Roch theorem

• arithmetic Riemann-Roch theorem

Related concepts

• Mordell conjecture

• arithmetic Chow group

• Vojta's conjecture

• global analytic geometry

• function field analogy

References

Introductions and surveys

• Wikipedia: Arakelov theory

• Serge Lang, Introduction to Arakelov theory Springer-Verlag, New York, 1988.

• Christophe Soulé, D. Abramovich, , J.-F. Burnol, J. Kramer, Lectures on Arakelov Geometry, Cambridge University Press 1991

• Robin de Jong, Explicit Arakelov geometry, PhD thesis 2004 (pdf)

• Alberto Camara, Notes on Arakelov theory, 2011 (pdf)

Original articles

The theory originates in

• Suren Arakelov, Intersection theory of divisors on an arithmetic surface, Math. USSR Izv. 8 (6): 1167–1180, 1974, doi; Theory of intersections on an arithmetic surface, Proc. ICM Vancouver 1975, vol. 1, 405–408, Amer. Math. Soc. 1975, djvu, pdf

After Arakelov there were main improvements by Faltings and Gillet and Soulé.

• Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387–424, MR86e:14009, doi; Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983), no. 3, 337–347, MR85m:14061, doi

The arithmetic Riemann-Roch theorem is due to

• Henri Gillet, Christophe Soulé, An arithmetic Riemann–Roch Theorem, Invent. Math. 110: 473–543, 1992, doi

• Shou-Wu Zhang, Small points and Arakelov theory, Proc. ICM 1998, vol. 2, djvu, pdf

In a recent Bonn thesis under Faltings’ supervision,

• Nikolai Durov, A new approach to Arakelov geometry, arxiv/0704.2030

a completely algebraic replacement (using generalized schemes whose local models are spectra of commutative algebraic monads) for the original mixed approach is proposed; it is not known if that approach can be closely and precisely compared with the traditional.

See also

• Jean-Benoit Bost, Klaus Künnemann, Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers (arXiv:math/0701343)

• Jean-Benoit Bost, Klaus Künnemann, Hermitian vector bundles and extension groups on arithmetic schemes. II. The arithmetic Atiyah extension (arXiv:0807.4374)

Relation to AdS3/CFT2

Relation of AdS3/CFT2 to hyperbolic geometry and Arakelov geometry of algebraic curves:

• Yuri Manin, Matilde Marcolli, Holography principle and arithmetic of algebraic curves, Adv. Theor. Math. Phys. 5 (2002) 617-650 (arXiv:hep-th/0201036