This book has been aimed at the first step in the following two-step program for the Riemann hypothesis:

1. (i) Work out Connes' approach in the geometric case of a curve over a finite field.

2. (ii) Translate this approach to a number field, the arithmetic case.

The second step naturally divides into two steps:

1. (a) Translate the local theory (archimedean and nonarchimedean case).

2. (b) Translate the global theory.

There is no difficulty in the translation of Connes' approach to the p-adic completions of a number field (step (iia) in the nonarchimedean case). This can already be found in Haran's work [Har3] and gives the Weil distribution as in Section 4.3. It is remarkable that also the archimedean local trace formula can already be found in the work of Weil, Haran, and Connes. The global theory, on the other hand, is incomplete, both in Chapter 6 of this book, and in the work of Connes and Haran.

One may wonder how much of the work towards a proof of the Riemann hypothesis will be accomplished with step (i). Is the geometric case essentially equivalent to the Riemann hypothesis? Or is the geometric case in a fundamental way simpler? One is reminded of the abc-conjecture, where the solution in the geometric case is fundamentally simpler and does not seem to give much insight for the arithmetic case.