Arakelov geometry

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) one, and "adding a point" $\infty$ to the corresponding 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 [a1], [a2] (for a restricted class of currents $g$). For the general theory, see [a7], [a9] and references therein.

Given a pair $( E , h )$ consisting of an algebraic vector bundle $E$ on $X$ and a $C ^ { \infty }$ 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 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 [a8], [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 [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 [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 [a10]. G. Faltings and E. Ullmo proved that $\omega ^ { 2 }$ is strictly positive [a4], [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 [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 [a5]: Assume $A$ is an 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

[a1]	S.J. Arakelov, "Intersection theory of divisors on an arithmetic surface" Math. USSR Izv. , 8 (1974) pp. 1167–1180 MR472815 Zbl 0355.14002
[a2]	S.J. Arakelov, "Theory of intersections on an arithmetic surface" , Proc. Internat. Congr. Mathematicians Vancouver , 1 , Amer. Math. Soc. (1975) pp. 405–408 MR466150
[a3]	J.-B. Bost, H. Gillet, C. Soulé, "Heights of projective varieties and positive Green forms" J. Amer. Math. Soc. , 7 (1994) pp. 903–1027 MR1260106 Zbl 0973.14013
[a4]	G. Faltings, "Calculus on arithmetic surfaces" Ann. of Math. , 119 (1984) pp. 387–424 MR0740897 Zbl 0559.14005
[a5]	G. Faltings, "Diophantine approximation on Abelian varieties" Ann. of Math. , 133 (1991) pp. 549–576 MR1109353 Zbl 0734.14007
[a6]	G. Faltings, "Lectures on the arithmetic Riemann–Roch theorem" Ann. Math. Study , 127 (1992) (Notes by S. Zhang) MR1158661 Zbl 0744.14016
[a7]	H. Gillet, C. Soulé, "Arithmetic intersection theory" Publ. Math. IHES , 72 (1990) pp. 94–174 MR1087394 Zbl 0741.14012
[a8]	H. Gillet, C. Soulé, "An arithmetic Riemann–Roch Theorem" Invent. Math. , 110 (1992) pp. 473–543 MR1189489 Zbl 0777.14008
[a9]	C. Soulé, D. Abramovich, J.-F. Burnol, J. Kramer, "Lectures on Arakelov geometry" , Studies Adv. Math. , 33 , Cambridge Univ. Press (1992) MR1208731 Zbl 0812.14015
[a10]	L. Szpiro, "Séminaire sur les pinceaux de courbes elliptiques (à la recherche de Mordell effectif)" Astérisque , 183 (1990)
[a11]	E. Ullmo, "Positivité et discrétion des points algébriques des courbes" Ann. of Math. , 147 : 1 (1998) pp. 167–179 MR1609514 Zbl 0934.14013
[a12]	P. Vojta, "Siegel's theorem in the compact case" Ann. of Math. , 133 (1991) pp. 509–548 MR1109352 Zbl 0774.14019