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''Arakelov theory''

A combination of the Grothendieck [[Algebraic geometry|algebraic geometry]] of schemes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202401.png" /> with Hermitian complex geometry on their set of complex points. The goal is to provide a geometric framework for the study of Diophantine problems in higher dimension (cf. also [[Diophantine equations, solvability problem of|Diophantine equations, solvability problem of]]; [[Diophantine problems of additive type|Diophantine problems of additive type]]).

The construction relies upon the analogy between number fields and function fields: the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202402.png" /> has Krull dimension (cf. [[Dimension|Dimension]]) one, and  "adding a point"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202403.png" /> to the corresponding [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202404.png" /> makes it look like a complete curve. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202405.png" /> is a rational number, the identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202406.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202407.png" /> is the valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202408.png" /> at the prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a1202409.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024010.png" />, is similar to the Cauchy residue formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024011.png" /></td> </tr></table>

for the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024012.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024013.png" /> is a non-zero rational function on a smooth complex projective curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024014.png" />.

In higher dimension, given a regular projective flat scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024015.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024016.png" />, one considers pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024017.png" /> consisting of an algebraic cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024018.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024019.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024020.png" />, together with a Green current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024022.png" /> on the complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024023.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024024.png" /> is real current of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024025.png" /> such that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024026.png" /> denotes the current given by integration on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024027.png" />, the following equality of currents holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024029.png" /> is a smooth form of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024030.png" />. Equivalence classes of such pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024031.png" /> form the arithmetic Chow group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024032.png" />, which has good functoriality properties and is equipped with a graded intersection product, at least after tensoring it by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024033.png" />.

These notions were first introduced for arithmetic surfaces, i.e. models of curves over number fields [[#References|[a1]]], [[#References|[a2]]] (for a restricted class of currents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024034.png" />). For the general theory, see [[#References|[a7]]], [[#References|[a9]]] and references therein.

Given a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024035.png" /> consisting of an algebraic [[Vector bundle|vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024037.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024038.png" /> [[Hermitian metric|Hermitian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024039.png" /> on the corresponding holomorphic vector bundle on the complex-analytic manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024040.png" />, one can define characteristic classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024041.png" /> with values in the arithmetic Chow groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024042.png" />. For instance, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024043.png" /> has rank one, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024044.png" /> is a non-zero rational section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024046.png" /> its divisor, the first [[Chern class|Chern class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024047.png" /> is the class of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024048.png" />. The main result of the theory is the arithmetic Riemann–Roch theorem, which computes the behaviour of the Chern character under direct image [[#References|[a8]]], [[#References|[a6]]]. Its strongest version involves regularized determinants of Laplace operators and the proof requires hard analytic work, due to J.-M. Bismut and others.

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024049.png" />, the pairings

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024050.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024051.png" />, give rise to arithmetic intersection numbers, which are real numbers when their geometric counterparts are integers. Examples of such real numbers are the heights of points and subvarieties, for which Arakelov geometry provides a useful framework [[#References|[a3]]].

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024052.png" /> is a semi-stable arithmetic surface, an important invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024053.png" /> is the self-intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024054.png" /> of the relative dualizing sheaf equipped with the Arakelov metric [[#References|[a1]]]. L. Szpiro and A.N. Parshin have shown that a good upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024055.png" /> would lead to an effective version of the Mordell conjecture and to a solution of the abc conjecture [[#References|[a10]]]. G. Faltings and E. Ullmo proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024056.png" /> is strictly positive [[#References|[a4]]], [[#References|[a11]]]; this implies that the set of algebraic points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024057.png" /> is discrete in its Jacobian for the topology given by the Néron–Tate height.

P. Vojta used Arakelov geometry to give a new proof of the Mordell conjecture [[#References|[a12]]], by adapting the method of Diophantine approximation. More generally, Faltings obtained by Vojta's method a proof of a conjecture of S. Lang on Abelian varieties [[#References|[a5]]]: Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024058.png" /> is an [[Abelian variety|Abelian variety]] over a number field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024059.png" /> be a proper closed subvariety in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024060.png" />; then the set of rational points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024061.png" /> is contained in the union of finitely many translates of Abelian proper subvarieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120240/a12024062.png" />.

See also [[Diophantine geometry|Diophantine geometry]]; [[Height, in Diophantine geometry|Height, in Diophantine geometry]]; [[Mordell conjecture|Mordell conjecture]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.J. Arakelov,   "Intersection theory of divisors on an arithmetic surface"  ''Math. USSR Izv.'' , '''8'''  (1974)  pp. 1167–1180</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.J. Arakelov,   "Theory of intersections on an arithmetic surface" , ''Proc. Internat. Congr. Mathematicians Vancouver'' , '''1''' , Amer. Math. Soc.  (1975)  pp. 405–408</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.-B. Bost,   H. Gillet,   C. Soulé,   "Heights of projective varieties and positive Green forms"  ''J. Amer. Math. Soc.'' , '''7'''  (1994)  pp. 903–1027</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Faltings,   "Calculus on arithmetic surfaces"  ''Ann. of Math.'' , '''119'''  (1984)  pp. 387–424</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Faltings,   "Diophantine approximation on Abelian varieties"  ''Ann. of Math.'' , '''133'''  (1991)  pp. 549–576</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Faltings,   "Lectures on the arithmetic Riemann–Roch theorem"  ''Ann. Math. Study'' , '''127'''  (1992)  (Notes by S. Zhang)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Gillet,   C. Soulé,   "Arithmetic intersection theory"  ''Publ. Math. IHES'' , '''72'''  (1990)  pp. 94–174</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H. Gillet,   C. Soulé,   "An arithmetic Riemann–Roch Theorem"  ''Invent. Math.'' , '''110'''  (1992)  pp. 473–543</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  C. Soulé,   D. Abramovich,   J.-F. Burnol,   J. Kramer,   "Lectures on Arakelov geometry" , ''Studies Adv. Math.'' , '''33''' , Cambridge Univ. Press  (1992)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  L. Szpiro,   "Séminaire sur les pinceaux de courbes elliptiques (à la recherche de Mordell effectif)"  ''Astérisque'' , '''183'''  (1990)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  E. Ullmo,   "Positivité et discrétion des points algébriques des courbes"  ''Ann. of Math.'' , '''147''' :  1  (1998)  pp. 167–179</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  P. Vojta,   "Siegel's theorem in the compact case"  ''Ann. of Math.'' , '''133'''  (1991)  pp. 509–548</TD></TR></table>