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In mathematics, Arakelov theory (or Arakelov geometry) is an approach to
Diophantine geometry In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...
, named for
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 ...
. It is used to study Diophantine equations in higher dimensions.


Background

The main motivation behind Arakelov geometry is the fact there is a correspondence between prime ideals \mathfrak \in \text(\mathbb) and finite places v_p : \mathbb^* \to \mathbb, but there also exists a place at infinity v_\infty, given by the Archimedean valuation, which doesn't have a corresponding prime ideal. Arakelov geometry gives a technique for compactifying \text(\mathbb) into a
complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
\overline 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 A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
\mathfrak of relative dimension 1 over \text(\mathcal_K) such that it extends to a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
X_\infty = \mathfrak(\mathbb) for every valuation at infinity. In addition, he equips these Riemann surfaces with
Hermitian metric In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on ea ...
s on
holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
s 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 In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism :X \times Y \to Y is a closed map (i.e. maps closed sets onto closed sets). Thi ...
. Note that other techniques exist for constructing a complete space extending \text(\mathbb), which is the basis of F1 geometry.


Original definition of divisors

Let K be a field, \mathcal_K its ring of integers, and X a genus g curve over K with a non-singular model \mathfrak \to \text(\mathcal_K), called an arithmetic surface. Also, we let \infty: K \to \mathbb be an inclusion of fields (which is supposed to represent a place at infinity). Also, we will let X_\infty be the associated Riemann surface from the base change to \mathbb. Using this data, we can define a c-divisor as a formal linear combination D = \sum_i k_i C_i + \sum_\infty \lambda_\infty X_\infty where C_i is an irreducible closed subset of \mathfrak of codimension 1, k_i \in \mathbb, and \lambda_\infty \in \mathbb, and the sum \sum_ represents the sum over every real embedding of K \to \mathbb and over one embedding for each pair of complex embeddings K \to \mathbb. The set of c-divisors forms a group \text_c(\mathfrak).


Results

defined an
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
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. extended Arakelov's work by establishing results such as a Riemann-Roch theorem, a
Noether formula Noether is the family name of several mathematicians (particularly, the Noether family), and the name given to some of their mathematical contributions: * Max Noether (1844–1921), father of Emmy and Fritz Noether, and discoverer of: ** Noether in ...
, a Hodge index theorem and the nonnegativity of the self-intersection of the dualizing sheaf in this context. Arakelov theory was used by
Paul Vojta Paul Alan Vojta (born September 30, 1957) is an American mathematician, known for his work in number theory on Diophantine geometry and Diophantine approximation. Contributions In formulating Vojta's conjecture, he pointed out the possible exis ...
(1991) to give a new proof of the
Mordell conjecture Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction. Educati ...
, and by in his proof of Serge Lang's generalization of the Mordell conjecture. developed a more general framework to define the intersection pairing defined on an arithmetic surface over the spectrum of a ring of integers by Arakelov. Arakelov's theory was generalized by
Henri Gillet Henri Antoine Gillet (born 8 July 1953, Tangier) is a European-American mathematician, specializing in arithmetic geometry and algebraic geometry. Education and career Gillet received in 1974 his bachelor's degree from King’s College London and ...
and
Christophe Soulé Christophe Soulé (born 1951) is a French mathematician working in arithmetic geometry. Education Soulé started his studies in 1970 at École Normale Supérieure in Paris. He completed his Ph.D. at the University of Paris in 1979 under the sup ...
to higher dimensions. That is, Gillet and Soulé defined an intersection pairing on an arithmetic variety. One of the main results of Gillet and Soulé is the arithmetic Riemann–Roch theorem of , an extension of the
Grothendieck–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is ...
to arithmetic varieties. For this one defines arithmetic
Chow group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ( ...
s CH''p''(''X'') of an arithmetic variety ''X'', and defines
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
es for Hermitian vector bundles over ''X'' taking values in the arithmetic Chow groups. The arithmetic Riemann–Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. A complete proof of this theorem was only published recently by Gillet, Rössler and Soulé. Arakelov's intersection theory for arithmetic surfaces was developed further by . The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space L^2_1. In this context, Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.


Arithmetic Chow groups

An arithmetic cycle of codimension ''p'' is a pair (''Z'', ''g'') where ''Z'' ∈ ''Z''''p''(''X'') is a ''p''-cycle on ''X'' and ''g'' is a Green current for ''Z'', a higher-dimensional generalization of a Green function. The arithmetic Chow group \widehat_p(X) of codimension ''p'' is the quotient of this group by the subgroup generated by certain "trivial" cycles.Manin & Panchishkin (2008) pp.400–401


The arithmetic Riemann–Roch theorem

The usual
Grothendieck–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is ...
describes how the
Chern character In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
ch behaves under pushforward of sheaves, and states that ch(''f''*(''E''))= ''f''*(ch(E)Td''X''/''Y''), where ''f'' is a proper morphism from ''X'' to ''Y'' and ''E'' is a vector bundle over ''f''. The arithmetic Riemann–Roch theorem is similar, except that the
Todd class In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encounte ...
gets multiplied by a certain power series. The arithmetic Riemann–Roch theorem states \hat(f_*( )=f_*(\hat(E)\widehat^R(T_)) where *''X'' and ''Y'' are regular projective arithmetic schemes. *''f'' is a smooth proper map from ''X'' to ''Y'' *''E'' is an arithmetic vector bundle over ''X''. *\hat is the arithmetic Chern character. *TX/Y is the relative tangent bundle *\hat is the arithmetic Todd class *\hat^R(E) is \hat(E)(1-\epsilon(R(E))) *''R''(''X'') is the additive characteristic class associated to the formal power series \sum_ \frac\left \zeta'(-m) + \zeta(-m) \left( + + \cdots + \right)\right


See also

*
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 . It bears the name of two mathematicians, Su ...
* Hodge theory *
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 ...
*
Adelic group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...


Notes


References

* * * * * * * * * * * * * *{{citation, first=Paul , last=Vojta, title=Siegel's Theorem in the Compact Case , journal=Annals of Mathematics , volume= 133, issue= 3 , year= 1991, pages= 509–548 , doi=10.2307/2944318, publisher=Annals of Mathematics, Vol. 133, No. 3 , jstor=2944318


External links


Original paperArakelov geometry preprint archive
Algebraic geometry Diophantine geometry