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Old 2011-02-19, 06:33   #1
Feb 2011

210 Posts
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
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fivemack's Avatar
Feb 2006
Cambridge, England

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The Weil conjectures refer to the zeta functions of algebraic varieties over finite fields, as defined at

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