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二番煎じでも、先の定理を拡張したり、一般化すれば、評価はまた変わる
Weil conjecturesの”second proof”

https://en.wikipedia.org/wiki/Weil_conjectures
Weil conjectures
(抜粋)
Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps:

Use of Lefschetz pencils
・The theory of monodromy of Lefschetz pencils, introduced for complex varieties (and ordinary cohomology) by Lefschetz (1924), and extended by Grothendieck (1972) and Deligne & Katz (1973) to l-adic cohomology, relates the cohomology of V to that of its fibers.
The relation depends on the space Ex of vanishing cycles, the subspace of the cohomology Hd?1(Vx) of a non-singular fiber Vx, spanned by classes that vanish on singular fibers.
・The Leray spectral sequence relates the middle cohomology group of V to the cohomology of the fiber and base.
c(U,E), where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibers the spaces Ex of vanishing cycles.

The key estimate
The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks.
This is done by studying the zeta functions of the even powers Ek of E and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups.
The crucial idea of considering even k powers of E was inspired by the paper Rankin (1939), who used a similar idea with k=2 for bounding the Ramanujan tau function. Langlands (1970, section 8) pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture,
and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization.
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