Deligne's work on the Weil conjectures

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?

fivemack · 2011-02-19, 11:30

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.

2011-02-19, 06:33	#1
intrigued Feb 2011 2₁₀ 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?

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2011-02-19, 11:30	#2
fivemack (loop (#_fork)) Feb 2006 Cambridge, England 7×911 Posts	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.