In memory of Maurice Auslander

In 1978 I gave a talk at the ENS in Paris. The title was “Faisceaux arithmétiques coherents”. My goal was to introduce the mathematical public to what I called “Arakelov theory”. To justify this introduction I explained what could be done with the idea of Arakelov and Parshin: “Put metrics at infinity on vector bundles and you will have a geometric intuition of compact varieties to help you”. I also explained that my seminar [Sz 1] written in geometric language, could be considered as a book of conjectures once one knew the translation of effective divisor, Kodaira-vanishing theorem, bounded families, Hodge index etc… Needless to say I did not raise enthusiasm at this point!

I present here the work that has been done on this program.

Faisceaux arithmétiques cohérents.

Heights.

Let K be a number field. The local-global equality defining the height of a point x ∈ Pn(K) is: (with L := O{1))

where Ex is the section of Pn OK → Spec Ok corresponding to the point x. This formula teaches us many things:

(1) It is a Riemann-Roch theorem in dimension one analogous to x(L) = deg L − g + 1 on a curve.

(2)The height is the “intersection” of a scheme of dimension 1 with a cycle of codimension 1. The fundamental theorem on heights is typical of the type of results one is able to get about Q:

Theorem 1 (Northcott's theorem).

Given d ∈ N and A ∈ R+ the set

As a corollary, once one knows h (the Neron-Tate height on an abelian variety A) one gets the finitness of the torsion of A(K) because (h(P) = 0 ≤> p has torsion).