  In a previous paper, we have defined arithmetic extension groups in the
context of Arakelov geometry. In the present one, we introduce an arithmetic
analogue of the Atiyah extension that defines an element -- the arithmetic
Atiyah class -- in a suitable arithmetic extension group. If $\overline{E}$ is
a hermitian vector bundle on an arithmetic scheme $X$, its arithmetic Atiyah
class is an obstruction to the algebraicity of the unitary connection on the
vector bundle $E_\C$ over the complex manifold $X(\C)$ that is compatible with
its holomorphic structure.
  We develop basic properties of the arithmetic Atiyah class and study its
vanishing in the case of hermitian line bundles. This may be translated into a
concrete problem of diophantine geometry, concerning rational points of the
universal vector extension of the Picard variety of $X$. We investigate this
problem, which was already considered and solved in some cases by Bertrand, by
using a classical transcendence result of Schneider-Lang, and we derive a
finiteness result.
  We also consider a geometric analog of our arithmetic situation, namely a
smooth, projective variety $X$ which is fibered on a curve $C$ defined over
some field $k$ of characteristic zero. To any line bundle $L$ over $X$ is
attached its relative Atiyah class ${\rm at}_{X/C}L$. We describe precisely
when this class vanishes. In particular, when the fixed part of the relative
Picard variety of $X$ over $C$ is trivial, this holds only when the restriction
of $L$ to the generic fiber $X_K$ of $X$ over $C$ is a torsion line bundle.

Algebraic Geometry;Number Theory