Arakelov theory

Last updated May 14, 2024

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

The main motivation behind Arakelov geometry is the fact there is a correspondence between prime ideals ${\mathfrak {p}}\in {\text{Spec}}(\mathbb {Z} )$ and finite places $v_{p}:\mathbb {Q} ^{*}\to \mathbb {R}$ , 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 ${\text{Spec}}(\mathbb {Z} )$ into a complete space

{\overline {{\text{Spec}}(\mathbb {Z} )}}

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 ${\mathfrak {X}}$ of relative dimension 1 over ${\text{Spec}}({\mathcal {O}}_{K})$ such that it extends to a Riemann surface $X_{\infty }={\mathfrak {X}}(\mathbb {C} )$ 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 ${\text{Spec}}(\mathbb {Z} )$ , which is the basis of F₁ geometry.

Original definition of divisors

Let $K$ be a field, ${\mathcal {O}}_{K}$ its ring of integers, and $X$ a genus $g$ curve over $K$ with a non-singular model ${\mathfrak {X}}\to {\text{Spec}}({\mathcal {O}}_{K})$ , called an arithmetic surface . Also, we let

\infty :K\to \mathbb {C}

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 $\mathbb {C}$ . 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 ${\mathfrak {X}}$ of codimension 1, $k_{i}\in \mathbb {Z}$ , and $\lambda _{\infty }\in \mathbb {R}$ , and the sum

\sum _{\infty }

represents the sum over every real embedding of $K\to \mathbb {C}$ and over one embedding for each pair of complex embeddings $K\to \mathbb {C}$ . The set of c-divisors forms a group ${\text{Div}}_{c}({\mathfrak {X}})$ .

Results

Arakelov ( 1974 , 1975 ) 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. GerdFaltings ( 1984 ) 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 GerdFaltings ( 1991 ) in his proof of Serge Lang's generalization of the Mordell conjecture.

PierreDeligne ( 1987 ) 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. Shou-WuZhang ( 1992 ) developed a theory of positive line bundles and proved a Nakai–Moishezon type theorem for arithmetic surfaces. Further developments in the theory of positive line bundles by Zhang ( 1993 , 1995a , 1995b ) and LucienSzpiro , Emmanuel Ullmo ,andZhang ( 1997 ) culminated in a proof of the Bogomolov conjecture by Ullmo ( 1998 ) and Zhang ( 1998 ).^[1]

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 Gillet & Soulé (1992), 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 Jean-BenoîtBost ( 1999 ). The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space $L_{1}^{2}$ . 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 ${\widehat {\mathrm {CH} }}_{p}(X)$ of codimension p is the quotient of this group by the subgroup generated by certain "trivial" cycles.^[2]

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 {\mathrm {ch} }}(f_{*}([E]))=f_{*}({\hat {\mathrm {ch} }}(E){\widehat {\mathrm {Td} }}^{R}(T_{X/Y}))

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.
${\hat {\mathrm {ch} }}$ is the arithmetic Chern character.
T_X/Y is the relative tangent bundle
${\hat {\mathrm {Td} }}$ is the arithmetic Todd class
${\hat {\mathrm {Td} }}^{R}(E)$ is ${\hat {\mathrm {Td} }}(E)(1-\epsilon (R(E)))$
R(X) is the additive characteristic class associated to the formal power series $\sum _{m>0 \atop m{\text{ odd}}}{\frac {X^{m}}{m!}}\left[2\zeta '(-m)+\zeta (-m)\left({1 \over 1}+{1 \over 2}+\cdots +{1 \over m}\right)\right].$

Notes

↑ Leong, Y. K. (July–December 2018). "Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry" (PDF). Imprints. No. 32. The Institute for Mathematical Sciences, National University of Singapore. pp. 32–36. Retrieved 5 May 2019.
↑ Manin & Panchishkin (2008) pp.400–401

Related Research Articles

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory.

The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

In mathematics, a modular form is a (complex) analytic function on the upper half-plane, $, that satisfies:$

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space $over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .$

The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space $, that is, n -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables, which the Mathematics Subject Classification has as a top-level heading.$

In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.

In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions P_ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces.

In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:

In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein. A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.

In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.

<span class="mw-page-title-main">Grothendieck–Riemann–Roch theorem</span>

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 itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.

In differential geometry, a discipline within mathematics, a distribution on a manifold $is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle .$

In algebraic geometry, a moduli space of (algebraic) curves is a geometric space whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.

In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F₁, or, in a French–English pun, F_un. The name "field with one element" and the notation F₁ are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F₁ refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of F₁ have been proposed, but it is not clear which, if any, of them give F₁ all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.

In mathematics, an arithmetic surface over a Dedekind domain R with fraction field $is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes. When R is the ring of integers Z, this intuition depends on the prime ideal spectrum Spec(Z) being seen as analogous to a line. Arithmetic surfaces arise naturally in diophantine geometry, when an algebraic curve defined over K is thought of as having reductions over the fields R / P, where P is a prime ideal of R, for almost all P; and are helpful in specifying what should happen about the process of reducing to R / P when the most naive way fails to make sense.$

This is a glossary of algebraic geometry.

Henri Antoine Gillet is an American mathematician, specializing in arithmetic geometry and algebraic geometry.

References

Arakelov, Suren J. (1974), "Intersection theory of divisors on an arithmetic surface", Math. USSR Izv., 8 (6): 1167–1180, doi:10.1070/IM1974v008n06ABEH002141, Zbl 0355.14002
Arakelov, Suren J. (1975), "Theory of intersections on an arithmetic surface", Proc. Internat. Congr. Mathematicians Vancouver, vol. 1, Amer. Math. Soc., pp. 405–408, Zbl 0351.14003
Bost, Jean-Benoît (1999), "Potential theory and Lefschetz theorems for arithmetic surfaces" (PDF), Annales Scientifiques de l'École Normale Supérieure , Série 4, 32 (2): 241–312, doi:10.1016/s0012-9593(99)80015-9, ISSN 0012-9593, Zbl 0931.14014
Deligne, P. (1987), "Le déterminant de la cohomologie", Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985)[The determinant of the cohomology], Contemporary Mathematics, vol. 67, Providence, RI: American Mathematical Society, pp. 93–177, doi:10.1090/conm/067/902592, MR 0902592
Faltings, Gerd (1984), "Calculus on Arithmetic Surfaces", Annals of Mathematics , Second Series, 119 (2): 387–424, doi:10.2307/2007043, JSTOR 2007043
Faltings, Gerd (1991), "Diophantine Approximation on Abelian Varieties", Annals of Mathematics , Second Series, 133 (3): 549–576, doi:10.2307/2944319, JSTOR 2944319
Faltings, Gerd (1992), Lectures on the arithmetic Riemann–Roch theorem, Annals of Mathematics Studies, vol. 127, Princeton, NJ: Princeton University Press, doi:10.1515/9781400882472, ISBN 0-691-08771-7, MR 1158661
Gillet, Henri; Soulé, Christophe (1992), "An arithmetic Riemann–Roch Theorem", Inventiones Mathematicae , 110: 473–543, doi:10.1007/BF01231343
Kawaguchi, Shu; Moriwaki, Atsushi; Yamaki, Kazuhiko (2002), "Introduction to Arakelov geometry", Algebraic geometry in East Asia (Kyoto, 2001), River Edge, NJ: World Sci. Publ., pp. 1–74, doi:10.1142/9789812705105_0001, ISBN 978-981-238-265-8, MR 2030448
Lang, Serge (1988), Introduction to Arakelov theory, New York: Springer-Verlag, doi:10.1007/978-1-4612-1031-3, ISBN 0-387-96793-1, MR 0969124, Zbl 0667.14001
Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.
Soulé, Christophe (2001) [1994], "Arakelov theory", Encyclopedia of Mathematics , EMS Press
Soulé, C.; with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer (1992), Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge: Cambridge University Press, pp. viii+177, doi:10.1017/CBO9780511623950, ISBN 0-521-41669-8, MR 1208731
Szpiro, Lucien; Ullmo, Emmanuel; Zhang, Shou-Wu (1997), "Equirépartition des petits points", Inventiones Mathematicae, 127 (2): 337–347, Bibcode:1997InMat.127..337S, doi:10.1007/s002220050123, S2CID 119668209 .
Ullmo, Emmanuel (1998), "Positivité et Discrétion des Points Algébriques des Courbes", Annals of Mathematics , 147 (1): 167–179, arXiv: alg-geom/9606017 , doi:10.2307/120987, Zbl 0934.14013
Vojta, Paul (1991), "Siegel's Theorem in the Compact Case", Annals of Mathematics, 133 (3), Annals of Mathematics, Vol. 133, No. 3: 509–548, doi:10.2307/2944318, JSTOR 2944318
Zhang, Shou-Wu (1992), "Positive line bundles on arithmetic surfaces", Annals of Mathematics , 136 (3): 569–587, doi:10.2307/2946601 .
Zhang, Shou-Wu (1993), "Admissible pairing on a curve", Inventiones Mathematicae , 112 (1): 421–432, Bibcode:1993InMat.112..171Z, doi:10.1007/BF01232429, S2CID 120229374 .
Zhang, Shou-Wu (1995a), "Small points and adelic metrics", Journal of Algebraic Geometry, 8 (1): 281–300.
Zhang, Shou-Wu (1995b), "Positive line bundles on arithmetic varieties", Journal of the American Mathematical Society , 136 (3): 187–221, doi: 10.1090/S0894-0347-1995-1254133-7 .
Zhang, Shou-Wu (1996), "Heights and reductions of semi-stable varieties", Compositio Mathematica , 104 (1): 77–105.
Zhang, Shou-Wu (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986, JSTOR 120986 .

External links

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Leong, Y. K. (July–December 2018). "Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry" (PDF). Imprints. No. 32. The Institute for Mathematical Sciences, National University of Singapore. pp. 32–36. Retrieved 5 May 2019.

[2] Manin & Panchishkin (2008) pp.400–401

[1]

[2]