Gerd Faltings

Calculus on arithmetic surfaces

In [A2] and [A3], Arakelov introduces an intersection calculus for arithmetic surfaces, that is, for stable models of curves over a number field. In this paper we intend to show that his intersection product has a lot of useful properties. More precisely, we show that the following properties from the theory of algebraic surfaces have an analogue in our situation:

The General Case of S. Lang's Conjecture

Publisher Summary This chapter discusses the general case of S. Langs conjecture. It presents the generalization of methods to yield all of S. Langs conjecture about rational points on subvarieties of Abelian varieties. A symmetric ample line-bundle is chosen and used to compute degrees. By noetherian induction, it can be assumed that Y is irreducible and then shown that these functions are constant on an open subset of Y . In the process, one is always allowed to remove a closed proper subset from Y . In general, after removing a proper closed subset from Y , X may be replaced by the union of the closures of the irreducible components of the generic fiber that are not contained in Z . Passing to a finite flat cover of Y , it may be assumed that all fibers are geometrically irreducible and not contained in Z . the bounds are allowed to increase in each step.

Finiteness Theorems for Abelian Varieties over Number Fields

Let K be a finite extension of ℚ, A an abelian variety defined over K, π = Gal(K/K) the absolute Galois group of K, and l a prime number. Then π acts on the (so-called) Tate module

Crystalline cohomology and GL(2, ℚ)

{T_l}(A) = \mathop{{\lim }}\limits_{{\mathop{ \leftarrow }\limits_n }} \,A[{l^n}](\overline K )

Coverings of p-adic period domains

The goal of this chapter is to give a proof of the following results: (a) The representation of π on \( {T_l}(A){ \otimes_{{{\mathbb{Z}_l}}}}{\mathbb{Q}_l} \) s is semisimple. (b) The map

F-Isocrystals on Open Varieties Results and Conjectures

{\text{En}}{{\text{d}}_K}(A){ \otimes_{\mathbb{Z}}}{\mathbb{Z}_l} \to {\text{En}}{{\text{d}}_{\pi }}({T_l}(A))