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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.
The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields. The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.
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.
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).
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.
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.
Hasse (1936) discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and Deuring (1941) developed their basic theory.
The Riemann zeta function is an example of an L-function, and some important conjectures involving L-functions are the Riemann hypothesis and its generalizations. The theory of L -functions has become a very substantial, and still largely conjectural , part of contemporary analytic number theory .