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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 ├ętale 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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