Weil Conjectures - Deligne's Second Proof

Deligne's Second Proof

Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem. Much of the second proof is a rearrangement of the ideas of his first proof. The main extra idea needed is an argument closely related to the theorem of Hadamard and de la Vallée Poussin, used by Deligne to show that various L-series do not have zeros with real part 1.

A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N(x)β/2, and is called mixed of weight ≤β if it can be written as repeated extensions by pure sheaves with weights ≤β.

Deligne's theorem states that if f is a morphism of schemes of finite type over a finite field, then Rif_! takes mixed sheaves of weight ≤β to mixed sheaves of weight ≤β+i.

The original Weil conjectures follow by taking f to be a morphism from a smooth projective variety to a point and considering the constant sheaf Q_l on the variety. This gives an upper bound on the absolute values of the eigenvalues of Frobenius, and Poincaré duality then shows that this is also a lower bound.

In general Rif_! does not take pure sheaves to pure sheaves. However it does when a suitable form of Poincaré duality holds, for example if f is smooth and proper, or if one works with perverse sheaves rather than sheaves as in Beilinson, Bernstein & Deligne (1982).

Inspired by the work of Witten (1982) on Morse theory, Laumon (1987) found another proof, using Deligne's l-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. Kiehl & Weissauer (2001) used Laumon's proof as the basis for their exposition of Deligne's theorem. Katz (2001) gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. Kedlaya (2006) gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology.