Etale Cohomology and Rationality of L-series of Varieties

Abstract

Etale cohomology was introduced by A. Grothendieck and developed by him with the help of M. Artin and J. L. Verdier, to explain A. Weil's insight that for polynomial equations with integer coefficients, the topology of the set of complex solutions should profoundly influence the number of solutions of the equations modulo a prime number. Weil conjectured that the zeta function (introduced by Hasse for curves and by Weil in general) of a smooth projective variety over a finite field $\mathbb{F}_q$, which is a generating function that captures the growth of the number of points defined over $\mathbb{F}_{q^n}$ as $n$ increases, is a rational function, satisfies a certain functional equation and has its zeroes at restricted places. Since \'etale cohomology gives a replacement, for arbitrary schemes, of the cohomology of the space of complex points of a variety, and since it gives a sheaf theory and cohomology theory whose properties closely resemble those arising from the complex topology, it allows for the use of topological ideas over general fields, both in algebraic geometry and in many areas (number theory, representation theory, algebra) where algebraic geometry plays an essential role.