Arakelov theory

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

Arakelov geometry studies a scheme "X" over the ring of integers Z, by putting 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.

Results

The theory was introduced for surfaces by harvs|authorlink=Suren Arakelov|last=Arakelov|year1=1974|year2=1975 and was used by Paul Vojta (1991) to give a new proof of the Mordell conjecture, and by harvs|txt=yes|first=Gerd |last=Faltings|authorlink=Gerd Faltings|year=1991 in his proof of Lang's generalization of the Mordell conjecture.

One of the main results of Arakelov theory is the arithmetic Riemann-Roch theorem of harvtxt|Gillet|Soulé|1992, an extension of the Grothendieck-Riemann-Roch theorem to arithmetic schemes. 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.

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 $hat\{CH\}\_p(X)$ of codimension "p" is the quotient of this group by the subgroup generated by certain "trivial" cycles.

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:$hat\{ch\}(f\_*(\; [E]\; )=f\_*(hat\{ch\}(E)hat\{Td\}^R(T\_\{X/Y\}))$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".
*$hat\{ch\}$ is the arithmetic Chern character.
*T_X/Y is the relative tangent bundle
*$hat\{Td\}$ is the arithmetic Todd class
*$hat\{Td\}^R(E)$ is $hat\{Td\}(E)(1-epsilon(R(E)))$
*"R"("X") is the additive characteristic class associated to the formal power series:$sum\_\{m>0,\; odd\}\{X^mover\; m!\}(2zeta^prime(-m)+zeta(-m)(\{1over\; 1\}+\{1over\; 2\}+cdots+\{1over\; m\})).$

References

*citation|first=S.J.|last= Arakelov|title=Intersection theory of divisors on an arithmetic surface|journal= Math. USSR Izv. |volume= 8 |year=1974|pages= 1167–1180 |doi=10.1070/IM1974v008n06ABEH002141
*citation|first=S.J.|last= Arakelov|chapter=Theory of intersections on an arithmetic surface|title= Proc. Internat. Congr. Mathematicians Vancouver |volume= 1 |publisher= Amer. Math. Soc. |year=1975|pages= 405–408
*citation|title=Calculus on Arithmetic Surfaces
first=Gerd|last= Faltings
journal=The Annals of Mathematics > 2nd Ser.|volume= 119|issue= 2 |year= 1984|pages= 387-424
url= http://links.jstor.org/sici?sici=0003-486X%28198403%292%3A119%3A2%3C387%3ACOAS%3E2.0.CO%3B2-X
*citation|title=Diophantine Approximation on Abelian Varieties
first=Gerd |last=Faltings
journal=The Annals of Mathematics > 2nd Ser.|volume= 133|issue= 3 |year= 1991|pages= 549-576
url= http://links.jstor.org/sici?sici=0003-486X%28199105%292%3A133%3A3%3C549%3ADAOAV%3E2.0.CO%3B2-X
*citation|id=MR|1158661
last=Faltings|first= Gerd
title=Lectures on the arithmetic Riemann-Roch theorem.
series= Annals of Mathematics Studies|volume= 127|publisher= Princeton University Press|place= Princeton, NJ |year=1992| ISBN= 0-691-08771-7
*citation|first=H.|last= Gillet|first2= C. |last2=Soulé|title=An arithmetic Riemann–Roch Theorem|journal= Invent. Math. |volume= 110 |year=1992|pages= 473–543|doi=10.1007/BF01231343
*citation|id=MR|2030448
last=Kawaguchi|first= Shu|last2= Moriwaki|first2= Atsushi|last3= Yamaki|first3= Kazuhiko
chapter=Introduction to Arakelov geometry|title= Algebraic geometry in East Asia (Kyoto, 2001)|pages= 1--74, |publisher=World Sci. Publ.|place= River Edge, NJ|year= 2002
DOI=10.1142/9789812705105_0001
*citation|id=MR|0969124
last=Lang|first= Serge
title=Introduction to Arakelov theory|publisher= Springer-Verlag|place= New York|year= 1988| ISBN= 0-387-96793-1
*springer|id=A/a120240|first=Christophe |last=Soulé
*citation|id=MR|1208731
last=Soulé|first= C.
title=Lectures on Arakelov geometry
With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer
series= Cambridge Studies in Advanced Mathematics|volume= 33|publisher=Cambridge University Press|place= Cambridge|year= 1992|pages= viii+177 | ISBN= 0-521-41669-8
*citation|first=Paul |last=Vojta|title=Siegel's Theorem in the Compact Case
journal=The Annals of Mathematics |volume= 133|issue= 3 |year= 1991|pages= 509-548
url= http://links.jstor.org/sici?sici=0003-486X%28199105%292%3A133%3A3%3C509%3ASTITCC%3E2.0.CO%3B2-Y

External links

* [http://www.institut.math.jussieu.fr/~vmaillot/Arakelov/ Arakelov geometry preprint archive]