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intrigued · 2011-02-19, 06:33

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 étale cohomology theory, then how come the Riemann hypothesis is still an open problem?

2011-02-19, 06:33	#1
intrigued Feb 2011 2 Posts	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 étale cohomology theory, then how come the Riemann hypothesis is still an open problem?