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Old 2011-02-19, 06:33   #1
intrigued
 
Feb 2011

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Post Deligne's work on the Weil conjectures

For the Frobenius automorphism F, Grothendieck proved that the zeta function \zeta(s) is equivalent to
\zeta(s) = \frac{P_1(T)\ldots P_{2n-1}(T)}{P_0(T)\ldots P_{2n}(T)},
where the polynomial P_i(T) = \det(L-TF) on the L-adic cohomology group H^{i}. In his 1974 paper, Deligne proved that all zeros of P_i(T) lie on the critical line of complex numbers s with real part i/2, 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?
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Old 2011-02-19, 11:30   #2
fivemack
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The Weil conjectures refer to the zeta functions of algebraic varieties over finite fields, as defined at http://en.wikipedia.org/wiki/Weil_co...il_conjectures

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