Introduction: Motivations from Geometry

Summary

In chap. 0 we begin with geometrical motivations and introduction. We recall the analogies between geometry (curve X over a finite field F _q ) and arithmetic (number field K), and the two basic problems of arithmetic: the problem of the real primes and the problem of non-existence of a surface Spec O _K × Spec O _K (analogues to X × F _q X). We then give the “Weil philosophy”: the explicit sums of arithmetic are the intersection number of Frobenius divisors on the (non-existing, but see [Har6]) surface. This was never made explicit by Weil (and only was spelled out in [Har2]). The proof of the functional equation and the Riemann-Roch in arithmetic give the “Tate philosophy”: we are studying the action of the idele-class A _K ^*/K^* on the problematic space A/K^*. The important part of the ergodic action of K^* on the Adele A _K is encoded in the action of K^* on A _K /K. We then recall the author formula that connects these two philosophies ([Har2], [Harl]), giving the explicit sums in terms of the Fourier transform of the degree log¦x¦_p⁻¹.