brauer groups, tamagawa measures, and rational points

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INTRODUCTION
then given by
h L
(x) := ̂deg ( x ∗ L ) .
Here, ̂deg denotes the Arakelov degree of a hermitian line bundle over Spec.
It turns out that this coincides exactly with the naive height when one works
with X = P n, L = O(1), and the minimum metric which is defined by
∣ ‖l‖ min := min
l ∣∣∣
i=0, ... ,n
∣ .
X i
In general, h L
admits a fundamental finiteness property as soon as L is ample.
This is the starting point of arithmetic intersection theory, a fascinating theory
which we may only touch upon. We recall the main definitions and results
of the arithmetic intersection theory due to H. Gillet and C. Soulé [G/S 90,
G/S 92, S/A/B/K]. As an application, we consider the concept of a height
introduced by J.-B. Bost, H. Gillet, and C. Soulé [B/G/S]. This is a height
function for cycles of arbitrary dimension. For a 0-dimensional cycle, the
Bost-Gillet-Soule height coincides with the usual height function defined by the
hermitian line bundle.
Wecloseour overviewofarithmeticintersectiontheoryby a sectiononthe arithmetic
Hilbert-Samuel formula. We formulate the formula as Theorem IV.6.5.
Asaconsequence, weproveS. Zhangs’s[Zh95b, (5.2)]theorem ofsuccessiveminima.
Arithmetic intersection theory seems to be a very general framework. It is,
however, still not general enough. An obvious problem is that the naive height
on P n is not covered. This problem is of technical nature. To define the intersection
product of two hermitian line bundles (or, more correctly, of the
corresponding arithmetic cycles), Gillet and Soulé had to assume that all hermitian
metrics are smooth. Unfortunately, the minimum metric is continuous,
but not smooth.
A theory which is general enough to cover the Bost-Gillet-Soule height as well
as the naive height is provided by S. Zhang’s adelic intersection theory [Zh95a].
We give a definition for a version of the adelic Picard group ˜Pic(X ). Elements of
˜Pic(X ) are called integrably metrized invertible sheaves. Then, we construct
the corresponding adelic intersection product. As an application, we explain
the height function defined by an integrably metrized invertible sheaf.
In Chapter 5, we use adelic intersection theory to give a unified proof for two
equidistribution theorems for points of small height. On one hand, there is
the equidistribution theorem of L. Szpiro, E. Ullmo and S. Zhang [S/U/Z]
for small points on abelian varieties. On the other hand, the completely parallel
theorem for the naive height on projective space which was established