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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.
In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann , is a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields .
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry.
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.
Deligne's contribution was to supply the estimate of the eigenvalues of the Frobenius endomorphism, considered the geometric analogue of the Riemann hypothesis. It also led to a proof of the Lefschetz hyperplane theorem and the old and new estimates of the classical exponential sums, among other applications.
Riemann hypothesis: number theory: ⇐Generalized Riemann hypothesis⇐Grand Riemann hypothesis ⇔De Bruijn–Newman constant=0 ⇒density hypothesis, Lindelöf hypothesis See Hilbert–Pólya conjecture. For other Riemann hypotheses, see the Weil conjectures (now theorems). Bernhard Riemann: 24900 Ringel–Kotzig conjecture: graph theory
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).