Arakelov Geometry
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, 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 equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
s 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 In algebra, an absolute value (also called a valuation, magnitude, or norm, although "norm" usually refers to a specific kind of absolute value on a field) is a function which measures the "size" of elements in a field or integral domain. More pre ...
, which doesn't have a corresponding prime ideal. Arakelov geometry gives a technique for compactifying \text(\mathbb) into a complete space \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 \mathfrak of relative dimension 1 over \text(\mathcal_K) such that it extends to a Riemann surface X_\infty = \mathfrak(\mathbb) 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 X. 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 \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 In mathematics, an arithmetic surface over a Dedekind domain ''R'' with fraction field K is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes. When ''R'' is the ring of integers ' ...
. 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 on the
arithmetic surface In mathematics, an arithmetic surface over a Dedekind domain ''R'' with fraction field K is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes. When ''R'' is the ring of integers ' ...
s 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, 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 (1991) to give a new proof of the Mordell conjecture, and by in his proof of
Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...
'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 In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
of integers by Arakelov. Arakelov's theory was generalized by Henri Gillet and Christophe Soulé 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 to arithmetic varieties. For this one defines arithmetic Chow groups CH''p''(''X'') of an arithmetic variety ''X'', and defines Chern classes 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 describes how the Chern character 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 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 *
Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
* P-adic Hodge theory *
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'' ...


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