Let be an affine, smooth geometrically connected curve. Let be an -adic local system or lisse -sheaf. It is given as a continuous, f.d. representation of over .
Define the Hasse-Weil L-function (a priori this lies in ). Grothendieck’s cohomological formula essentially says that we can understand this -function with each Euler factor equal to the inverse of the characteristic polynomial of local Frobenius in terms of the characteristic polynomial of the global Frobenius acting on the compactly supported cohomology where is base change of to .
Because is affine, its compactly supported cohomology are concentrated in degree 1 and 2. The degree 2 is the dual of up to some shift.
The idea of Katz’s proof is that the moduli space of genus curves are path connected (using existence of space-filling curves over finite fields), and if is an affine curve connecting and where we know RH holds for , then we can hope to bootstrap and prove RH for . Say is the curve fibration where the two end points are and .
Fact: There is a closed relationship between and where is a closed point of . Say is defined over , then The fundamental compatibility is that
We want to prove the local system is pure of weight one. Replacing by the one half Tate-twisted local system on which is divided by , we see that it suffices to show all eigenvalues of any have, via any field embedding , absolute value (Because then it implies on itself, all eigenvalues of any have, via , absolute value ; and from the functional equation the inequality is an equality).
We already know it for one closed point (corresponding to ). The idea is to the cohomological expression of the -function (combined with the Rankin-Selberg trick!) implies that bounds on the absolute value of the image of the eigenvalues of on under can be used to bound that of the local Euler factor (The key condition on is -real, which implies for even tensor power , the local Euler factors are in ).
But for we have a description in terms of coinvariants. More precisely, and viewing these coinvariants as a quotient representation of, the action of is just the action of on this quotient. In other words, is among the eigenvalues of on .
Reference:
https://web.math.princeton.edu/~nmk/baby16.pdf
https://math.berkeley.edu/~fengt/Weil_I.pdf