A Concise Account of the Weil Conjectures and Etale Cohomology

The story of the Weil conjectures and the development of etale cohomology is the story of one of the great triumphs of 20th century algebraic geometry. The problem treated is exceedingly elementary – counting solutions of polynomials over finite fields – but the proofs require and indeed motivated the creation of an astonishing array of sophisticated technical machinery. The earliest echoes of the Weil conjectures may be found in two very different sources: Gauss’ work on counting solutions to polynomial equations modulo p, which arose in relation to Gauss sums; and Riemann’s study of the zeroes of the zeta function, leading to the Riemann hypothesis. These themes were brought together by E. Artin in his thesis in the 1920’s, when he developed an analogue of the zeta function associated to curves over finite fields, verified the Riemann hypothesis in some examples, and conjectured that it holds for all curves. Following further work of F. K. Schmidt on the form of the zeta function of curves over finite fields, and Hasse on the Riemann hypothesis for elliptic curves, Weil was able to conclude the story for curves by proving the Riemann hypothesis. It is interesting to note that it was precisely in order to make his arguments rigorous that Weil wrote Foundations of Algebraic Geometry, in which he introduced the notion of an abstract algebraic variety, obtained by gluing together affine varieties. In 1949, Weil went further, in what must surely be the most monumental paper to have appeared in the Bulletin of AMS. In it, Weil generalized what was known for curves, giving a precise conjecture as to the form of the zeta function for any smooth, projective variety over a finite field.