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''Arakelov theory''

A combination of the Grothendieck [[algebraic geometry]] of schemes over $\mathbf{Z}$ 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 problems of additive type]]).

The construction relies upon the analogy between number fields and function fields: the ring $\mathbf{Z}$ has Krull dimension (cf. [[Dimension|Dimension]]) one, and "adding a point" $\infty$ to the corresponding [[Scheme|scheme]] $\operatorname {Spec}( \mathbf{Z})$ makes it look like a complete curve. For instance, if $f \in \mathbf{Q} ^ { * }$ is a rational number, the identity

\begin{equation*} \sum _ { p } v _ { p } ( f ) \operatorname { log } ( p ) + v _ { \infty } ( f ) = 0, \end{equation*}

where $v _ { p } ( f )$ is the valuation of $f$ at the prime $p$ and where $v _ { \infty } ( f ) = - \operatorname { log } | f |$, is similar to the Cauchy residue formula

\begin{equation*} \sum _ { x \in C } v _ { x } ( f ) = 0 \end{equation*}

for the differential $d f / f$, when $f$ is a non-zero rational function on a smooth complex projective curve $C$.

In higher dimension, given a regular projective flat scheme $X$ over $\mathbf{Z}$, one considers pairs $( Z , g )$ consisting of an algebraic cycle $Z$ of codimension $p$ over $X$, together with a Green current $g$ for $Z$ on the complex manifold $X ( \mathbf{C} )$: $g$ is real current of type $( p - 1 , p - 1 )$ such that, if $\delta _{\text{Z}}$ denotes the current given by integration on $Z ( \mathbf{C} )$, the following equality of currents holds:

\begin{equation*} d d ^ { c } g + \delta _ { Z } = \omega, \end{equation*}

where $\omega$ is a smooth form of type $( p , p )$. Equivalence classes of such pairs $( Z , g )$ form the arithmetic Chow group $\widehat { \operatorname {CH} } ^ { p } ( X )$, which has good functoriality properties and is equipped with a graded intersection product, at least after tensoring it by $\mathbf{Q}$.

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 $g$). For the general theory, see [[#References|[a7]]], [[#References|[a9]]] and references therein.

Given a pair $( E , h )$ consisting of an algebraic [[Vector bundle|vector bundle]] $E$ on $X$ and a $C ^ { \infty }$ [[Hermitian metric|Hermitian metric]] $h$ on the corresponding holomorphic vector bundle on the complex-analytic manifold $X ( \mathbf{C} )$, one can define characteristic classes of $( E , h )$ with values in the arithmetic Chow groups of $X$. For instance, when $E$ has rank one, if $s$ is a non-zero rational section of $E$ and $\operatorname { div } ( s )$ its divisor, the first [[Chern class|Chern class]] of $( E , h )$ is the class of the pair $( Z , g ) = ( \operatorname { div } ( s ) , - \operatorname { log } ( h ( s , s ) ) )$. 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 $\widehat { CH } ^ { 1 } ( \operatorname { Spec } ( \mathbf{Z} ) ) = \mathbf{R}$, the pairings
$$
\widehat { CH }^p (X) \otimes \widehat { CH } ^ {n-p}(X) \rightarrow \widehat { CH } ^ { 1 } ( \operatorname { Spec } ( \mathbf{Z} ) ) \,,
$$
$p \geq 0$, 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 $X$ is a semi-stable arithmetic surface, an important invariant of $X$ is the self-intersection $\omega ^ { 2 }$ 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 $\omega ^ { 2 }$ 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 $\omega ^ { 2 }$ is strictly positive [[#References|[a4]]], [[#References|[a11]]]; this implies that the set of algebraic points of $X$ 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 $A$ is an [[Abelian variety|Abelian variety]] over a number field and let $X \subset A$ be a proper closed subvariety in $A$; then the set of rational points of $X$ is contained in the union of finitely many translates of Abelian proper subvarieties of $A$.

See also [[Diophantine geometry]]; [[Height, in Diophantine geometry]]; [[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 {{MR|472815}} {{ZBL|0355.14002}} </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 {{MR|466150}} {{ZBL|}} </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 {{MR|1260106}} {{ZBL|0973.14013}} </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 {{MR|0740897}} {{ZBL|0559.14005}} </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 {{MR|1109353}} {{ZBL|0734.14007}} </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) {{MR|1158661}} {{ZBL|0744.14016}} </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 {{MR|1087394}} {{ZBL|0741.14012}} </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 {{MR|1189489}} {{ZBL|0777.14008}} </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) {{MR|1208731}} {{ZBL|0812.14015}} </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 {{MR|1609514}} {{ZBL|0934.14013}} </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 {{MR|1109352}} {{ZBL|0774.14019}} </td></tr>
</table>