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The Riemann Hypothesis for Function Fields by Machiel van Frankenhuijsen

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7

Epilogue

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

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

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

The second step naturally divides into two steps:

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

(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 ...

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