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The proof of the Riemann hypothesis for varieties over finite fields by Deligne (1974) is possibly the single strongest theoretical reason in favor of the Riemann hypothesis. This provides some evidence for the more general conjecture that all zeta functions associated with automorphic forms satisfy a Riemann hypothesis, which includes the ...
However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by Deligne , using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.
Deligne's 1980 paper contains a much more general version of the Riemann hypothesis. From 1970 until 1984, Deligne was a permanent member of the IHÉS staff. During this time he did much important work outside of his work on algebraic geometry.
Specifically, the Riemann Hypothesis is about when 𝜁(s)=0; the official statement is, “Every nontrivial zero of the Riemann zeta function has real part 1/2.” On the plane of complex numbers ...
The last two parts were quite consciously modeled on the Riemann zeta function and Riemann hypothesis. The rationality was proved by Dwork (1960), the functional equation by Grothendieck (1965), and the analogue of the Riemann hypothesis was proved by Deligne (1974).
Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of the conjectures in 1960 using p-adic methods), and the remaining conjecture, the analogue of the Riemann hypothesis was proved by Pierre Deligne (1974) using ℓ-adic cohomology.
The extended Riemann hypothesis asserts that for every number field K and every complex number s with ζ K (s) = 0: if the real part of s is between 0 and 1, then it is in fact 1/2. The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q, with ring of integers Z.
Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see Kleiman (1968).