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-   -   Deligne's work on the Weil conjectures (https://www.mersenneforum.org/showthread.php?t=15274)

 intrigued 2011-02-19 06:33

Deligne's work on the Weil conjectures

For the Frobenius automorphism [TEX]F[/TEX], Grothendieck proved that the zeta function [TEX]\zeta(s)[/TEX] is equivalent to
[TEX]\zeta(s) = \frac{P_1(T)\ldots P_{2n-1}(T)}{P_0(T)\ldots P_{2n}(T)},[/TEX]
where the polynomial [TEX]P_i(T) = \det(L-TF)[/TEX] on the L-adic cohomology group [TEX]H^{i}[/TEX]. In his [URL="http://tinyurl.com/62shqva"]1974 paper[/URL], Deligne proved that all zeros of [TEX]P_i(T)[/TEX] lie on the critical line of complex numbers [TEX]s[/TEX] with real part [TEX]i/2[/TEX], a geometric analogue of the Riemann hypothesis.

My question is that if Deligne proved the Riemann hypothesis using étale cohomology theory, then how come the Riemann hypothesis is still an open problem?

 fivemack 2011-02-19 11:30

The Weil conjectures refer to the zeta functions [b]of algebraic varieties over finite fields[/b], as defined at [url]http://en.wikipedia.org/wiki/Weil_conjectures#Statement_of_the_Weil_conjectures[/url]

These were called zeta functions by analogy with the Riemann zeta function, but they're not the same thing.

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